Leonardian Fluid Mechanics in the Codex Atlanticus Iv-V

LEONARDIAN FLUID MECHANICS IN THE CODEX ATLANTICUS IV-V

By Enzo Macagno

IIHR Monograph No. 107

Iowa Institute of Hydraulic Research The University of Iowa Iowa City, Iowa 52242-1585

March 1989 H .H.C.

The University of Iowa Libraries

presented by ENZO MACAGNO QC142 L46M252 1989 LEONARDIAN FLUID MECHANICS IN THE CODEX ATLANTICUS IV - V

By ENZO MACAGNO

Sponsored by National Science Foundation

and National Endowment for the Humanities

IIHR Monograph No. 107 Iowa Institute of Hydraulic Research The University of Iowa Iowa City, Iowa 52242-1585 March 1989

i SE VOJ FARE PRUOVA QUALE PARTE DACQA DEL UASO CHE VERSA HE QUELLA CHE VIENE FUORJ FA TALE SPERIENTIA CHOL UASO DEL MJGLIO IL QUALE E LUBRICHO E MJNVTO E STOPA E DISTOPPA VARI BUSI DI TALE VASO E VEDERAJ SEL PIANO DI SOPRA DEL MJGLIO CHALA DI SE QUELLA PARTE CHE STA PERPENDICHULARE SOPRA DELL USSCITA DI SOTTO ONNO.

E SSE TTU DICIESSI QUESTA NON ESSERE BONA SPRERIENTIA PERCHE LACQUA IN SE QUANTJTA VNJTA E CONTINVA EL MJGLIO E DISUNJTO E DISCONTINVO A CQUESTA PARTE IO TI RISPONDO CHE IO VO PIGLIARE QUELLA LICIENZA CHE CHOMVNE AI MATEMATJCJ COE SICCHOME LORO DIVIDANO IL TENPO A GRADI E DI QUANTITÀ CONTINVA LA FANNO DISCONTINVA AÑORA IO FARO IL SIMJLE DANDO COL MJGLIO O RENELLA CONPARATIONE ALL ACQUA. Codex Arundel 349R

ii TABLE OF CONTENTS

INTRODUCTION Opening Remarks...... 1 Leonardo's Most Original Work...... 2 THE CODEX ATLANTICUS Brief History...... 4

Publications of the Codex Atlanticus...... 5 METHODOLOGY USED IN THIS STUDY

Identification of Passages of Interest...... 8 Organization of this Volume...... 10 LEONARDO’S METHODOLOGY Innovative Work Always Results from a Methodology...... 12 Analogy, Paradox and Experimentation...... 12

THE QUESTION OF THE ADEQUATE CRITERION...... 15

FUTURE W ORK...... 19

ACKNOWLEDGEMENTS...... 20 REFERENCES...... 21 CODEX ATLANTICUS, SECTION IV...... 1-35 CODEX ATLANTICUS, SECTION V...... 1-47 LEONARDIAN FLUID MECHANICS IN THE CODEX ATLANTICUS

INTRODUCTION

Opening Remarks

In Monographs 100 and 105 , the reader will find introductory comments for this work on the Codex Atlanticus. To give to each volume unity and a selfcontained style, I am repeating some of the basic notions and concepts which guide me in this work. But there are also new thoughts and views because, as time passes, I perceive new aspects that were not seen a year, or even a few months, ago. One of the fascinating thrills that this work has brought to me is that I feel as an explorer in the vast territories covered by Leonardo in his studies of just one science. I feel more enthusiastic than ever, and I end almost every day with something new for my mind to think about. Somewhere, Leonardo says that one should use the time before sleep comes to think about some of the great wonders before us. In a way, I do that, but thinking about the area of his writings and drawings I have explored during the day. I never thought that I was engaging in such a rewarding kind of work.

In this introduction, the reader will find first a reference to my assessment of the work of Leonardo on fluids and their properties, and on flow and transport phenomena. Then I describe briefly the Codex Atlanticus as a document now contained in twelve large volumes, with the drawings and handwritten texts of Leonardo, and twelve regular size volumes with the diplomatic and critical transcriptions of the text. Following this, the question of the methodologies is presented. First,the methods I have developed and expanded for the analysis and synthesis of Leonardo's studies are presented and discussed, second, Leonardo's own methods and procedures, and third the important question of the adequate method of description for a synthetical view of Leonardian Fluid Mechanics.

1 Leonardo's Most Original Work Leonardo da Vinci was more original in his fluid-mechanical studies than in any other field of endeavor, as I have already explained in recent publications [Macagno 1985b, 1987a]. Over a period of about forty years, Leonardo studied many flow and transport phenomena using a methodology in which analogy, paradox, experiments, trial and error, and observation played a central role. Leonardian fluid mechanics has only been studied fragmentarily and sporadically in the past, without an effort being made at encompassing all of the extant notes and drawings concerning flow and transport phenomena of water, air, fire and granular materials (corresponding to 'acqua, aria, foco, terra' in Leonardo's terminology). I do not forget the excellent publications of F. Arredi and R. Giacomelli on

Leonardian hydrostatics and aeronautics respectively, but they only show that constructive critical work like theirs is rare in the long history of writings about the work of Leonardo

[Arredi 1942-43, Giacomelli 1936]. Although the work of Arredi and Giacomelli is very important, they could not have had the view of fluid mechanics that is possible today, after this science has acquired a new status due to the efforts that began with L. Prandtl in Germany and G. I. Taylor in England [See the collected works of this two great leading figures].

A synthesis of Leonardo's science of flow from the original documents remains to be arrived at, and it will require, no doubt, a monumental work if one takes into account the nature of the numerous documents, none of which contains a systematic compilation on flow phenomena. In this respect, the Codex Atlanticus is an example of complete lack of order in the topics; it is true that this is not really the fault of Leonardo but of the compiler of the codex. A synoptical preview of the great variety of topics covered by Leonardo in his studies of flow science is possible by examining IIHR Monograph No. 100[Macagno

1986b] as well as my publications on the Madrid, Hammer, Forster and Arundel Codices, and the Ms A France [Macagno 1982, 1985a, 1986a, 1987 c, d, 1988a,b]. More will be

2 available in the near future as outputs of my ongoing project concerning the French

Manuscripts. But not only the work itself is of great interest, one must consider also the methodology of Leonardo which is still more difficult to trace; on this aspect, I have already written some contributions [Macagno 1982, 1985a, 1985b]. In fact, one can wonder what is more interesting: whether the observation and analysis of so many different phenomena, or the innovative way in which they were studied by Leonardo.

3 THE CODEX ATLANTICUS

Brief History

According to Professor Carlo Pedretti [1978], we do not really have a codex, because a codex is "a book, the sheets of which are already sewn in signatures when its pages come to be filled with notes." What has been called the Codex Atlanticus is a collection of sheets which was put together by Pompeo Leoni, at the end of the sixteenth century. On the cover, Leoni put the inscription : "Disegni di machine et delle arti secreti et altre cose di Leonardo da Vinci raccolti da Pompeo Leoni." The present name comes from the atlas format. Pedretti reports that Baldassare Oltrocchi, librarian of the Biblioteca

Ambrosiana, called this volume "Codice delle sue carte informa Atlantica" and also "Codice Atlantico." This name is universally accepted, although it is misleading, because what we really have is a collection of papers which were mounted by Pompeo Leoni on the white sheets of a large book (65 cm x 44 cm).

According to Professor Augusto Marinoni [1954, 1975], the painter Pompeo Leoni came into possession of most of the notebooks and papers left by Leonardo da Vinci to

Francesco Melzi. Marinoni estimates that there were about fifty notebooks and nearly 2000 loose sheets. He does not believe (as Pedretti does) that Leoni destroyed or damaged notebooks to prepare his album, and mentions a study by André Corbeau [1968] in support of his opinion, based on the fact that there are no traces in the sheets to have been ever sewn into a notebook. Marinoni grants, however that Leoni used liberally the scissors and the knife in his separation of notes and drawings referring to artistic and anatomical subjects from those on technical matters. Leoni appears to have attempted to put in some logical order the sheets and their fragments, but was rather unsuccessful. An example of how Leoni messed up things is given by Marinoni in his discussion of folios 169, 173 and

177. Leoni did also some damage through his mounting of sheets on inappropriate windows cut in the blank pages of the original atlas. But we must still be thankful to Leoni

4 because he helped in preserving hundred of sheets for posterity. It is believed that about half of the papers of Leonardo, if not more, have been lost due to lack of interest and care.

One interesting facet of the history of the Codex Atlanticus is that it was not always highly valued as it is nowadays. Marinoni reports that, a few years after the death of Leoni in 1608, his son Giovan Battista offered the Codex to Cosimo II de’Medici together with fifteen smaller books by Leonardo. An expert, Giovan Francesco Cantagallina, an engineer, was consulted and produced a negative report, saying that the codex was very trivial, not worth of being possessed by the grand duke. Some years later, the Codex was acquired by the Milanese count Galeazzo Arconati from Vittoria Leoni, daughter of

Pompeo. It seems that Milano has honored always Leonardo more than Firenze did.

Arconati, after a few years, made a donation to the Biblioteca Ambrosiana of eleven manuscripts of Leonardo, including the Codex Atlanticus. There the Codex remained until

May 1796 when Napoleon decided to transfer it to Paris together with a number of notebooks of Leonardo. It was stated that the transfer was requested by G.B. Venturi, a man from Modena (Italy), who was a professor in Paris, and wanted to study the manuscripts. After the fall of Napoleon, the Austrian government requested from the

French government the restitution of the pillaged books, but only the Codex Atlanticus was returned to the Biblioteca Ambrosiana.

Pubiicati9ns--Qf...tìi€,-CQd£x .Atlanticus The manuscripts kept in Paris were published by Ravaisson-Mollien in between 1881 and 1891, about four centuries after Leonardo wrote them. This seems to have stimulated the Italians to do something alike with the once companion document in the

Biblioteca Ambrosiana, the Codex Atlanticus, and in between 1894 and 1904, under the editorial effort of Giovanni Piumati, Hoepli of Milano did the first publication of the Codex. The process for the second printing started in 1962, when the Codex was transferred from the Ambrosiana Library to the monastery of Grottaferrata to be restored

5 with funds provided by the Italian government. The restoration was accomplished under the direction of Father Giosafat Kurilo. According to Marinoni, the old volume, or album, is now at the Ambrosiana: "come spoglia di una larva mutata in un essere più splendido". Pedretti is much more critical of the work done in this restoration, which was completed in

1970.1 consider that I must mention these aspects, as well as others of scientific interest, because there is always a great danger of inflicting serious damage when-perhaps with the best intentions—somebody undertakes to introduce changes in the documents of the past. Pedretti summarizes his opinion in the following sentence: "It is in fact much to be regretted that the 'restoration has often resulted in serious damage to the originals." In his

Catalogue[1978] of the newly restored sheets, Pedretti points out for each folio the damage done, and the errors in mounting the sheets. He also gives the estimated date for each page, an information which has been used in this work.

The restored Codex Atlanticus of Leonardo da Vinci has been published by Giunti- Barbèra of Firenze [1975-80]. The editor was Prof. Augusto Marinoni of Milano. It consists of twelve large volumes with plates, and twelve smaller volumes with the transcriptions (diplomatic and critical) by Marinoni [1975]. There are 1119 folios, of which

998 contain only one of the original sheets; in facsimile, of course. The remaining folios contain from two up to thirteen sheets. The numbering of the folios in the two versions of the Codex Atlanticus is quite different. Marinoni has included in each volume the correspondence between old and new numbers. Such correspondence was also published in the new version of the book by F. Calvi [1925] on the manuscripts of Leonardo da Vinci, edited by Marinoni in 1982.

I do not envision that my contribution could be used, for any definitive work, independently from the facsimiles of the Codex Atlanticus. The two facsimiles are separated by nearly a century, and I have found the old one as useful as the new one. In a number of cases, the quality of the drawings has been better preserved in the old facsimile. I believe that my work can be useful to different studies focusing on areas of fluid

6 mechanics and transport processes in a much better way that any analytical index. I know this from experience because my survey and excerpts have been extremely useful in preparing my contribution on analogies in the Codex Atlanticus [Macagno 1986c]. The texts have been supplemented with all those words in Italian (in Leonardesque really) which have presented some difficulty. In this way any reader gets a warning about possible weak points in my version. The drawings have been in a number of cases simplified, and always subject to interpretation to incorporate my view of them as a fluid- mechanicist. They are no substitute for those of Leonardo. For any investigation, one should finally rely on the original drawings; but it is very useful to have easily at hand text and drawings together for all the preliminary steps of a study. Contrary to what I did with the texts, in the case of the drawings I have included only those which have a direct relation to fluid mechanics, or to questions of general mechanics which are important in fluid mechanics.

7 METHODOLOGY USED IN THIS WORK

Identification of Passages of Interest

Reading every word and examining every drawing of the Codex has been the basic approach adopted. One cannot do otherwise, because the risk of missing something important is there all the time. The content of each folio is totally unpredictable and, although part of the work does not yield anything, the reward for being totally careful is great anyway, because in many cases one finds valuable hidden jewels. Leonardo, as any other great figure of the past, cannot be studied using the so common standard of our schools: that one can earn partial credit. Any important work should either be done completely or not done at all. Those who do not read Leonardo's original writings and drawings, or examine them superficially, are surely doomed to do a very poor work.

Professor Marinoni has emphasized many times the need for reading the original. The same advice was given to me by the late Nando de Toni, a life-time student of Leonardo's notebooks. However, with all the respect the author has for these two great Italian

Leonardists, he is always careful to check their transcriptions against the facsimiles of Leonardo's notebooks when making notes of passages of interest. The author has endeavored not to make still more true Marinoni's comment on students of Leonardo:

"Leonardo è uno degli autori più celebrati ma purtroppo meno letti." (In a free translation:

"Many talk about Leonardo, few read him". ) Marinoni's criticism reaches many whose studies and writings on the works and thinking of Leonardo are based on anthologies, disregarding the thousands of original pages Leonardo left as a heritage, and although part of it has been lost, there.is much that has not been properly explored and presented yet. I have described my methodology in IIHR Monograph 100 [Macagno 1986b] and in other publications; however, it seems warranted to say a few words about it here. Each time a passage was found which dealt with fluids behavior, or flow phenomena, or topics

8 related with fluid mechanics, notes were taken, and then a number of words--the

keywords-were selected, first tentatively, and, in a second sweep, definitively. This approach is intended to detect the notions and concepts and I try to represent each finding with a profile containing as many words as necessary, while still maintaining low the total number of terms used. This procedure is radically different from that used in indices of words. At the beginning of the survey, the same set of keywords that were useful in representing the content of the Madrid Codices was used [Macagno 1982], but gradually more and more words became necessary. The procedure went through a phase of trial and error, of attempts or "pentimenti " (to use a Leonardian expression for trial and error).

What Leonardo did in his artistic drawings [see Gombrich 1966] was done with each of his sketches on flow phenomena, in each of his folios. There was no attempt to draw a perfect sketch the first time one was needed, but the effort was repeated as many times as necessary to be satisfied that it was accomplished. Leonardo himself explained how one must work to achieve something, just like a writer who tries a first draft, and then a second, and maybe, even a third one before being satisfied. Thus, for each folio a set of terms and/or a drawing was determined and then entered into a table. This tabulation is not the end of this representation. As different topics are considered, multiple-channel tables can be formed as the one already completed for the analogies in this Codex and included in a recent publication [Macagno 1986c]. Once the above preliminary study was completed, I undertook the work which appears in this volume.

In this volume, I have gathered all the passages that I found relating directly or indirectly to fluid mechanics in the folios 271 to 452 of the Codex Atlanticus of Leonardo da Vinci. There are a total of 1119 folios, and I will publish several volumes more in the near future for the remaining 667 folios. I have rendered into English the notes of interest,

and I have drawn myself sketches based on those of Leonardo. In this way a version of both the written and the pictographic records on flow science resulted and it is offered

9 because I consider it essential for an ulterior rational synthesis, by myself and others, of Leonardian fluid mechanics.

Organization of this Volume

In this monograph the reader will find a double page arrangement, whenever drawings have been included to illustrate the passages transcribed from the Codex Atlanticus. The page on the left-hand side contains my version in English of such passages, and my indication of words or sentences that caused me some problems. The words in question are underlined, and the corresponding words in Italian (or more precisely, in Leonardesque) are given. On the right-hand side page there is a succinct "map" of the corresponding page in the codex, showing the approximate location of the drawings. The reader should remember that almost any map involves some distortion. Also included are the corresponding drawings; I have drawn the sketches myself, because I want to give my own interpretation of them as a fluid-mechanicist.

As I have explained already in other publications, I have resigned myself to the frustrating task of translating into English Leonardo's notes on the science of flow. I do this with some unhappiness because I do not have full command of the English language.

But then, I see others with absolutely no knowledge of the science of flow trying to put into English what Leonardo wrote. And I try to compensate for the damage they do. The ideal approach would be to learn what Leonardo wrote by learning his own language; which is what I have done through many years of painstaking effort. But then, how is this work on Leonardian fluid mechanics to be completed? Surely not in the years I have left to live. Others will have to do it. And the chances are that such efforts will come from English-speaking researchers supported by funds from English-speaking countries. My dearest hopes are that there will be some researchers who will make the effort of learning enough of the original language. There is a fortunate circumstance, however, and this is is that much of the science of flow is the form of pictographic rather than written language.

10 In many cases, I have gained perhaps more insight from the drawings than from the text.

In using this volume (as well as IIHR Monograph 105), one should take into account IIHR Monograph 100 which contains the profiles for all the folios of interest.In this respect, this volume differs from the IIHR Monographs for Mss H and C. Another difference is that I have not reproduced the Italian text in this case. I regret this, but it would have represented a tremendous amount of work for which I have no time and

energy. One could say that, there are transcriptions available, but they do not follow my

approach of trying to transcribe with a structuring of words with as much as possible respect for the way in which Leonardo wrote but still giving a word by word transcription.

Leonardo did not separate words well and one must break many arbitrary combinations of letters all the time and put together the words he meant to wright. I have adopted an eclectic approach to translation, because each passage needs to be rendered as a unique piece. In some cases, I thought it was better to be as literal as

possible, to convey what Leonardo was clearly saying; in other cases, I found that he was

not too careful with his writing, but his intention was transparent, and I treated those

passages as something to be explained,.as a matter of exegetical translation. In addition to

this, I have had no hesitation in showing my doubts whenever they existed. It is perhaps paradoxical, but it seems that the more one knows about a given field, the more difficult it seems to be sure of what Leonardo says concerning specific topics in such a field. Only generalists seem to make confident translators and not to be disturbed by doubts. In documents as those left by Leonardo there are many ambiguities and many obscure passages. It is quite probable that some points may remain for ever in doubt. There is still much work to be done before we can offer for the general public a truly coherent

synthetical version of all the fluid mechanical work of Leonardo. However, consistent

views in some areas are emerging and I would like to refer the reader to the papers I have

already published (see References).

11 LEONARDO'S METHODOLOGY

Innovative Work Always Results from a Methodology

Did Leonardo possess, or developed, a methodology of his own? He surely enounces some important guiding principles and criteria concerning the use of mathematics in his studies, or the recourse to experimentation. He does it in some cases in unequivocal terms, even in a boisterous form. Bur how can these chaotic notes, many of these hurriedly done drawings, these disorganized records, reflect a methodology, and particularly, a scientific methodology? He did not have a university degree; how can one launch the notion that he had a methodology like the scholars of his time? My answer to these questions is simple, if one creates something new for enough time, in art, or science, or technology, or any human activity, one is at the same time creating his own method. It is quite possible that creators and inventors may be little aware of having a methodology. We know that powerful minds have pondered how were they able to create new knowledge (e.g., Hadamard 1945, Poincaré 1913, 1944). So Leonardo must have had his methodology not only as a painter, but as an anatomist, a fluid-mechanicists, etc. I have endeavored to trace such a methodology, and I have already written about my conclusions, which are still preliminary, but that I feel are an effort worth doing. I have followed in this respect, the leadership of Gombrich who undertook almost four decades ago to trace the method of Leonardo as an artist.

Analogy» Paradox and Experimentation Leonardo used a methodology in his studies of flow and transport phenomena in which experiments, analogies and paradoxes played roles of paramount importance. He was innovative, and well aware of being so, in a number of cases in which his involvement was much more profound than in questions of geometry, or fluid mechanics, or transport phenomena. I refer to his discussion, in his Treatise on Painting, on the use of

12 componimento inculto, as a manifestation of great awareness of how he thought that an artist should proceed in his creative work. This question has been masterfully discussed by Sir Ernst Gombrich (1952) in a paper on the methodology followed by Leonardo for working out his drawing and painting compositions. I think that the method in which pentimenti and componimento inculto were put to use appears very much like the trial-and- error procedure which has been and is so useful in science and technology. Of course, art was a field in which Leonardo must have felt much more at home than physics. I use the word physics here as comprising his notes and drawings on geometry, perspective, optics, mechanics, fluid mechanics, thermodynamics, transport phenomena. I consider that the really creative part of the geometrical writings and drawings of Leonardo belongs more to physics than to mathematics [Macagno M. 1987]. Leonardo ventured boldly into areas which he was not really prepared to enter by training, education, or his cultural background. He dared, and surely enjoyed, to act not as much as an engineer but as what we would consider today a research engineer, or an applied physicist. He did not have in these endeavors somebody to guide him, as he could have had in hydraulic engineering, for instance. In the case of geometry, if Luca Paccioli did really try to teach Leonardo some classic notions, it does not seem that Leonardo learned much; and if there would have been anybody around with some knowledge of physics and willing to teach him, I believe that he would only have had a negative influence on Leonardo. What happened, because Leonardo could not read Latin well enough to be imbued by the old arguments, was that he was much better off, and dared to challenge the old physics or, what is more important, to start developing a new physics. This was accomplished more by what he drew, as attested by many of his drawings, than by what he wrote down. I think that what was needed then, more than anything else, was to disregard or plainly ignore the static, frozen science of the past, which in the case of physics was full of errors and misconceptions. The great achievement of Leonardo in science appears to be that he was able to make some use of, and then go beyond the few notions of the mathematics and the physics of his time that he

13 was able to absorb. This is specially true in the case of flow and transport phenomena; but I believe that it may be found that this is true as well of other sciences in Leonardo's notebooks. I have chosen to open this volume with a quotation from the Codex Atlanticus (folio 349) which illustrates very well how Leonardo arrived at its own interpretation of ideas about continuum and discontinuum which probably floated around him. And he did not lose himself in pure speculation, but applied them to an experimental analogy in which the flow of water was going to be investigated by means of the flow of granular material. I have included two photographs of my own experiments at the University of Karlsruhe in which I studied the differences and the analogies between flow of water and the flow of sand. [Macagno 1975]

As I have already stated, I have come to the conclusion that the studies of Leonardo in physics were essentially the result of using analogical thinking, paradoxical argumentation, experimental methodology, and trial-and-error procedures. I know that such methods are old in mankind, but my thesis is that there is a very innovative use of them in Leonardo. Even when Leonardo repeats in his notes something already known for a long time (like the Archimedes' principle of hydrostatics) one can expect something new; to be sure, this newness is often marred by some misconception or failure to grasp properly the classic result, but it also carries some innovative way of understanding and of gaining understanding. Not many in the history of mankind have jumped from one plateau of knowledge to another in the long process of trying to improve our image of the physical world. We should not forget that this is done in great part by qualitative thinking rather than by quantification. In Leonardian fluid mechanics there is some quantification (particularly concerning conservation laws, in which he was quite advanced), but most of it is qualitative, as it should be as one goes from one plateau to another in the early developments of any science.

14 THE QUESTION OF THE ADEQUATE CRITERION

One aspect of the approach that I have followed seems to need more emphasis and more discussion; I think it is fully warranted to offer a few comments on it. I feel more and more that the influence of some historians of science, among which Neugebauer is a special and very representative case, has bom fruits that I was not so aware of just some time ago.

I am leaning very strongly towards the approach which includes experimentation and analysis based on the advanced knowledge of our times, in an effort to describe, especially in a synthesis, the work of the past under the light of what was going to be the future for any particular figure of centuries past. I do not berate, of course, the efforts to observe the case in question against the light from the respective past but doing only this is very short­ sighted.

To be sure,it is very important to know what Leonardo learnt from reading , or interacting with those who read, to know whatever direct or indirect sources existed for his inherited knowledge,to determine which of his notes reflect acquired knowledge and which ones reflect his own discoveries. For instance, one has much to learn from reading

Aristóteles’ Physics, and still more from the Problemata [See References]. The latter is specially reminiscent of a number of passages in Leonardo's notebooks. However, I believe that there has been too much work and speculation about who were or were not read by Leonardo, and about who read Leonardo after his time. He never wrote a treatise, all we have is like a disorganized, chaotic, incomplete diary. It is surely important to know which ones among the notes contain ideas coming from the past, but it is still much more important to detect what was new, original, innovative. I see an analogy with the analysis of X-ray pictures and other images; we can use a variety of forms of illumination. Let us not make the mistake of use only one kind of light.

For instance, Leonardo was obviously influenced by the old concepts of

Aristotelian physics, and to a certain extent by medieval physics.[See the excellent

15 publications of M. Clagett]. But if one does not know enough the physics of our times, how can one detect the generation of proto-ideas?. Let us say, that somebody feels that Leonardo was presenting some thoughts, however rudimentary, about non-Euclidean geometry, or about kinematics, or some hints of conservation principles, or some proto-

Newtonian concepts, how can one assess, and judge, and discriminate without a solid modern scientific background? It is time to do away with the notion that scholars can analyze historical developments of science and not know science well. If somebody believes that the only science is the logico-deductive one, how can one perceive all the innovative trends in Leonardo, or in any other figure of the past.? With such a criterion the brilliant contributions of the Mesopotamian astronomical science would be ignored[see the studies of Otto Neugebauer who took a truly scientific approach]. If one has not done experiments, how can one assess if a reference to experiments is illusory, or has some grounds? Unbelievable as they may seem, many have analyzed and pronounced judgement on Leonardian engineering and science who knew not enough science, and engineering, and had not real experience in any of those fields. It should be understood that we are bound to express and describe what Leonardo discovered in a very different manner than his own; I am afraid this tends to disturb those who are not familiar with recent developments in the study of the history of science which happen to require a great expertise in science, but I do not see other solution than undertaking an examination of the situation and considering the alternatives. Take, for instance, the great accomplishments of the astronomers of Mesopotamia and Greece; it is actually impossible to make a serious analysis of such achievements without using modem knowledge and language of mathematics and astronomy. Already in 1874, Schiaparelli published in Milano an analysis using advanced mathematics to examine the theory of homocentric spheres of Eudoxus. [For more recent studies see the writings of Otto Neugebauer, already mentioned above.] The same should apply to any study of

Leonardo's science; we would miss important fundamental points, were we not to use

16 modem language and modern points of view. The question of the adequate methodology is extremely important in the study of documents of the past, and undoubtedly a delicate one. Let us further clarify this question with examples. First, suppose that somebody has some idea that Leonardo introduced in his notebooks notions of non-Euclidean geometry, let us say some proto-concepts of elliptic geometry; how could anyone without a thorough knowledge of such a geometry examine the work of Leonardo to see the extent and the depth of his notions? The same is true for any other discipline. I must insist that there are aspects in the analysis of Leonardo's notes that can only be seen in their proper value when analyzed in the light of knowledge acquired after him rather than that existing before him. I would like to mention at this point that Greenberg [1980] has referred to the paradoxical fact that knowledge of non-Euclidean geometry is what really helps in understanding better Euclidean geometry. In the same way, knowledge of modem fluid mechanics is what helps best in understanding the historical development of hydraulics and hydromechanics [Macagno 1982].

As a second example, consider hydrostatics, which was well advanced already by

Archimedes; we must confront carefully what Leonardo wrote about the subject with Archimedes' work; but here again, if we do not use the deepest knowledge of hydrostatics available to us, our analysis will fail to be incisive. We can examine the work of Archimedes and capture the sense of his achievements because we understand pressure much better than he did. Archimedes grasped very well the notion of hydrostatic force and not too well that of pressure [Dijksterhuis 1957] Leonardo, instead, had serious misunderstandings concerning hydrostatic force[Macagno 1982] but he developed good insight about pressure [Macagno 1985a, 1987b]. We must take into account also that it is through rather simple questions that one can best trace Leonardo's methodology; because of this, my discussions do not touch upon some of the spectacular studies of Leonardo concerning flow and transport phenomena [Macagno 1982, 1985b, 1986a, 1987b], or upon machines and mechanisms, or hydraulic works. Knowing how advanced

17 engineering was in Lombardy, one wonders how much of the machines and hydraulic works in the notebooks of Leonardo is really original, and how much are in fact notes about existing engineering practice.

18 FUTURE WORK

Regarding the Codex Atlanticus it remains to put together the rest of the material selected in the same form as the first five of the twelve volumes of the second publication of this document. Then the work of synthesis can proceed. By the time this phase of the study of the Codex Atlanticus is completed, I hope to have about half of the Manuscripts at the Institute de France completed also. The Codices Hammer, Forster and Arundel have already being the object of similar analytical studies. It will remain to do a more systematic work for the Codices Madrid than the one I published in 1982, and the analysis of other documents with less fluid mechanical content. [See, e.g. Clark, 1968]. The work of synthesis has already been started and soon I will proceed at a more intense pace. I hope to attract collaborators for the work of synthesis in some areas which are less known to me than that of general fluid mechanics and hydraulics. Specially important is to find scholars interested in co-operating in the synthesis of fluvial hydraulics, flow machines, and some areas of physics.

19 ACKNOWLEDGEMENTS

My research work on Leonardian Fluid Mechanics in the Codex Atlanticus received financial support through a joint grant from National Science Foundation and the National Endowment for the Humanities. I am also grateful for a Fulbright Award for the academic year 1986-87, during which I was able to work with Professor Augusto Marinoni at the

Biblioteca d'Arte of the city of Milano, in Italy. The staff of the Biblioteca d'Arte has been extremely kind and helpful. I am greatly indebted to Professor Marinoni of the Commissione Vinciana of Italy for his advice and help, always granted with great generosity and efficiency. I am also grateful to the Politecnico of Milano, in which Istituto di Idraulica I have been given a spacious office and secretarial help.

Many persons have helped me by agreeing to discuss aspects of my work, or by lending a hand in libraries, laboratories and class rooms, or by kindly and efficiently answering my letters full of questions. In this respect, I extend my gratitude especially to Professors A. Marinoni and V. Vanoni. Many of my students have helped me unknowingly by answering quizzes taylored to discover primitive notions or their reactions to questions that were considered by Leonardo. They also performed experiments which were in some way repetitions of those of Leonardo. My wife has helped me in all possible ways, not the least being her constructive criticism blended with an unshakable faith in this work. Finally, Ms Karen Nall, of the Iowa Institute of Hydraulic Research, has assisted me very efficiently in the preparation of the manuscript for the printing office. Her kindness at all times is very much appreciated.

20 REFERENCES

ARISTOTLE, Physics. Translated by R. Hope, University of Nebraska Press, 1961. ARISTOTLE, The works of Aristotle , translated into English under the editorship of W.D.Ross. Oxford at Clarendon Press ,1927. See voi. VII, Problemata. by E.S. Forster. Particularly interesting are the sections on Mathematics, Salt water and the sea, Hot water, Air, Winds. The format in which questions are posed is reminiscent of Leonardo's similar passages; only that Leonardo leaves unanswered a number of questions. There is the possibility that he thought that the Aristotelian answer was not satisfactory..

ARREDI F. 1942-43. Le origini dell'idrostatica. L'Acqua, fase. 3, 7, 11, 12 of 1942; fase. 2, 5 of 1943.

CALVI, G. (1925) I manoscriti di Leonardo da Vinci dal punto di vista cronologico, storico e biografico. Bologna, Italy.

CALVI, G. (1982) I manoscriti di Leonardo da Vinci. Edited by A. Marinoni. Bramante Editrice, Italy.

CLAGETT, M. 1959 The Science of Mechanics in the Middle Ages. The University of Wisconsin Press, Madison, Wisconsin.

This is an important reference for the study of Leonardo under the light of the past. See also other volumes of the "Publications in Medieval Science" published by the University of Wisconsin., especially the book by Moody and Clagett, 1952. CLARK, K. (1968) Windsor Catalogue. Voi. I, Introduction. London.

CORBEAU, A. (1968) Les manuscrits de Léonardo de Vinci. Caen, France.

DIJKSTERHUIS.,EJ. 1957. Archimedes. The Humanities Press, New York. (There is a paperback printing by Princeton University, 1987.)

Excellent example of modem adequate methodology. Dijksterhuis was a professor of History of Mathematics and Natural Sciences at the University of Utrecht. See, on p. 379, the statement that our concept of pressure was alien to Archimedes. See also comments by W. R. Knorr at the end of the book.( p. 419) GIACOMELLI, R. 1936. Gli scritti di Leonardo da Vinci sul volo. Bardi. Roma

GOMBRICH, E.H., 1966. Norm and Form. Studies in the Art of the Renaissance. Phaidon Press, London.

See "Leonardo's Method of Working out Compositions", pp. 58-63. The plates 93 & 94 show the verso and recto of Study for the Virgin with St. Anne. Anybody who wants to understand Leonardo's trial-and-error methodology would do well to study carefully these two plates. In my opinion, this is a beautiful example of how the artist was armed with methods of universal application, enabling him to make progress in his scientific investigations.

21 GREENBERG, M.J. 1980. Euclidean and Non-Euclidean Geometries. W.H. Freeman, New York.

A theorem that is attributed to Leonardo is stated in a form similar to the following: In the Euclidean plane and in the hyperbolic plane, the only finite groups of motions are the cyclic and the dihedral groups of order n > 1. For a much more elementary presentation, which is however much longer, see WEYL 1952. HADAMARD, J. 1945. The Psychology of Invention in the Mathematical Field. Princeton University Press, Princeton.

This book is one of the most interesting essays I know about the exploration of creativity in the mathematical mind. Hadamard gives credit to Henri Poincaré for the inspiration to do his work. (See Appendices I and II for an interesting inquiry concerning the working methods of scientists, and Albert Einstein answer to it.) LEONARDO da VINCI, Il Codice Atlantico di Leonardo da Vinci. Pubi, by Reale Accademia dei Lincei. Hoepli, Milano, Italy, 1894-1904. LEONARDO DA VINCI. Il Codice Atlantico. Pubi, by Commissione Vinciana. Giunti Barbèra, Firenze, Italy, 1975-80. MACAGNO, E. 1975-85. Internal Reports of the Instituí für Hydromechanik Università Karlsruhe. Karlsruhe, West Germany.

MACAGNO, E. 1982. La meccanica dei fluidi nei Codici di Madrid, Scientia. Special volume. Milano, Italy.

A number of applications of the laboratory methodology to the study of Leonardo's notebooks are illustrated in this contribution. Some of the experiments are from the Codex Atlanticus.

MACAGNO, E. 1985a. Hidrostática Vinciana en el Códice Hammer. Anales de la Universidad de Chile. Quinta serie, No. 8.

In this contribution several remarkable experiments by Leonardo are studied, including some of the paradoxical situations which were probably used in the way of reductio ad absurdum.

MACAGNO, E. 1985b. Leonardo's Methodology in his Fluid Mechanical Investigations, Proceedings International Symposium on Modeling and Turbulence. Paper K3, IIHR, The University of Iowa, Iowa City, Iowa, USA.

MACAGNO, E., 1986a. What has not being explored in the Codex Hammer. Invited Lecture, Symposium on Leonardo in a new Perspective. Spencer Museum of Art, The University of Kansas, Lawrence, March 22, 1986. (A revised version has been published in 1988 as a volume of the series Leonardian Fluid Mechanics, IIHR Monograph No. 101, by The University of Iowa.)

MACAGNO, E. 1986b. Leonardian Fluid Mechanics in the Codex Atlanticus.A Survey. voi. I, IIHR Monograph No. 100, The University of Iowa, Iowa City, LA,USA. MACAGNO, E. 1986c. Analogies in Leonardo's Studies of Flow Phenomena. Studi Vinciani. Centro Ricerche Leonardiane, Brescia, Italy.

22 MACAGNO, E. 1987a. Leonardo da Vinci: Engineer and Scientist, in Hydraulics and Hydraulic Research, a Historical Review, ed. by G. Garbrecht, A.A. Balkema, Rotterdam. (Expanded version of the Berlin lecture in 1985). MACAGNO, E. 1987b. La noción de presión en la mecánica de fluidos Vinciana. Raccolta Vinciana. Fascicolo XXII, Milano, Castello Sforzesco

This paper is mainly about the paradox that Leonardo understood pressure better than hydrostatic force, but contains also considerations about the use by Leonardo of paradoxes in his study of hydrostatics.

MACAGNO, E. 1987c. Multichannel Tabulation of the Notes on Flow in the French Manuscripts of Leonardo da Vinci. Raccolta Vinciana. Fascicolo XXII. Castello Sforzesco,Milano, Italy MACAGNO, E. 1987d. Unexplored Flow Studies in the Codex Arundel 263. Internal Report, Istituto di Idraulica, Politecnico di Milano. Milano, Italy. (To be published as an IIHR Monograph.) MACAGNO, E. 1988a. Leonardian Fluid Mechanics. What Remains to be Investigated in the Codex Hammer. A Critical Study and a Challenge. IIHR Monograph No. 101. The University of Iowa, Iowa City, IA, USA.

MACAGNO E, 1988b. Leonardian Fluid Mechanics. Unexplored Flow Studies in the Codices Forster. IIHR Monograph No. 102. The University of Iowa, Iowa City, IA, USA.

MACAGNO, E. 1988c. Leonardian Fluid Mechanics in the Manuscript H. IIHR Monograph No. 103. The University of Iowa, Iowa City, IA, USA. MACAGNO,M. 1987. Geometry in Motion in the Manuscripts of Leonardo da Vinci. Internal Report Istituto di Idraulica Politecnico de Milano. MACAGNO, M, and E. MACAGNO, 1987. Geometrical Configurations in the Manuscripts of Leonardo da Vinci. Internal Report of the Istituto di Idraulica di Milano, Italy. (Scheduled for publication in Raccolta Vinciana).

MARINONI, A. (1954) I manoscriti di Leonardo da Vinci e le loro edizioni. In Leonardo. Saggi e ricerche. Roma, Italy.

MARINONI, A. (1975) Il Codice Atlantico della Biblioteca Ambrosiana di Milano. Volume Primo. Giunti-Barbèra, Firenze, Italy.

MOODY E.A. and CLAGETT, M. 1952. The Medieval Science of Weights. The University of Wisconsin Press, Madison, Wisconsin. Treatises ascribed to Euclid, Archimedes, Thabit ibn Qurra, Jordanus de Nemore and Blasius of Parma. Note that in the Middle Ages, statics was known as Scientia de Ponderibus (Science of Weights). PRANDTL, L. 1961. Gesammelte Abhandlungen zum angewandten Mechanik. Hvdro- und Aerodvnamik. Springer. Berlin.

23 I believe that if one is familiar with the Prandtlian approach to fluid flow phenomena, one has a view that is very adequate to the study of Leonardian science of flow and transport phenomena. The two fluiddynamicists could not have been more different, however there are important lines in common on these two men which would be very interesting to study in depth. To a certain extent, I believe that Sir Geoffrey Taylor offers also a similar possibility.[See Taylor 1958]

PEDRETTI, C. (1972) Leonardo da Vinci: The Rovai Palace of Romorantin. Cambridge, Mass.

PEDRETTI, C. (1977) Commentary to the Literary works of Leonardo da Vinci. Voi. II. London.

PEDRETTI, C. (1978) The Codex Atlanticus of Leonardo da Vinci. A Catalogue of its Newly Restored Sheets. Part one, Foreword and Introduction. Johnson Reprint Corporation & Harcourt Brace Jovanovich Publishers. Printed in Italy. POINCARÉ, H. 1913. The Foundations of Science. The Science Press, New York.

POINCARÉ, H. 1944. Ciencia v Método. Espasa-Calpe Argentina, Buenos Aires.

See Chapter III on invention in mathematics.

RAVAISSON-MOLLIEN, C. 1888. Les Manuscripts de Léonard de Vinci. Mss de la Biliothèque de l'Institut. Maison Quantin, Paris. SCHIAPARELLI, G.V., 1874. Die homocentrischen Sphàren des Eudoxus, Kallipus und Aristóteles. Memoire gelesen in Lombardischen Institut zu Mailand am 26 Nov. 1874. Translated by W.Horn and published in Abhandlungen zur Geschichte der Mathematik, erstes Heft, Leipzig, 1877.TAYLOR, G.I. 1958-1971. Scientific Papers. Cambridge University Press.

WEYL,H, 1952. Symmetry. Princeton University Press, Princeton, New Jersey. Excellent book, very well illustrated, written for the general public in a very attractive style by a mathematician with a genuine interest in art. See p. 66. More comments on the way Leonardo approached the preservation of symmetry are to be found on p. 99. It is interesting to see how Leonardo approached a problem which is now treated and described in an entirely different way, but which is recognized as being the same by the modern mathematician. Compare with the way in which GREENBERG 1980 formulates the findings of Leonardo. Theses references are important in connection with the adequate methodology and description in the analysis of documents of the past.

24 CODEX ATLANTICUS

SECTION IV IV -1

CA 280 R (102 r.b) c. 1503-5

The flux and reflux of the sea continually frusso erreflusso moves earth with all the elements (away) from the center of the elements. It is proven by the first of this, which says that the center prima di questo of the world takes into account what is lar­ ger than that center, because no 'conca' is less than the center. The center of the world is fixed by nature, but the place where per se it is moves all the time in different direc- tions. The place of the center of the world asspectj changes continually; such changes are some slower than others. Some take place each six hours but others happen with mi 11ion-of-years migliara intervals.

But the period of six hours originates in the flux and reflux of the sea; while the other derives from the erosion of the mountains due consumatione to the flow of water due to the rains and to the continuous course of the rivers. The place of the center change but not the center of the place, because such a center is fixed, and the place moves continually in straight motion, and such motion will never be curvi- linear. churvilinjo

CA 281 R (102 v.a) c. 1516-17

Knowledge of similitude in fluid mechanics is so important 3 that references to geometric IV-1 IV -2

similarity in Leonardo's notes are included in this work. The following lines were written close to a drawing with squares3 circles and triangles :

The ratio from circle to circle is the same prone as that of square to square constructed by the multiplication of the diameters of such circles by themselves, as it is proven in its in se medesimj place.

CA 282 R (102 v.b) c. 1500-5

Comments for a perpetuum-mohit e hydraulic wheel3 with the tetters rqponmLK, be d e f g h i, and a n m

'Soff issticho'.

The center of gravity of the waters r q p o n m L K be in n. Hence the arm n f contains in lieua it 12 times the counter-arm m n, because the contra lieua weiqht f must overcome the said counter- vjeere weights rqponmLK and the weight hr because it is over the center of the wheel in the vertical line, amounts to nothing in perpendichulare •potentia' of movement. i offers resistance to the descent of o, and h offers resistance to the descent of d, and g to the descent of e. Hence, only f remains against the said weight. IV-2 IV-3

CA 287 a R (103 v.c) c. 1505, or 1503-4?

Two sketches of jets coming out of reser­ voirs. In one case the water is led upon the blades of a hydraulic wheel with the letters g-onmcba-tSrqp.

All the water g o rests on the blade t, and pala the Tester n rests in part on the £>lad_e s dente and in part on the canal S t. Because of the ' 6a del quinto' m does not exert much pressure on the blade r. dente IV-3 IV -4

CA 299 R (108 r.a) c. 1508

Of motion and weight

In equal motions accomplished in equal time the 'potentia' of the motor will be greater motore than that of the moved body. And so much more powerful will the motor be in as much as the motion of the body exceeds the length of lungheza the motion of the motor. (Rut) so much less will be the difference between the ’potentia' of the motor and that of the moving body in as much less the motion accomplished by the moving body exceeds the motion of the such motor. But, understand, reader, that in this case account must be taken of the air. The air condenses more in front of the moving chondensa body the greater its velocity is; because the air is indefinitely condensable. This (com- inf initamente pressibility) does not play a role for mo­ tions of bodies moving in water, which is incompressible. This can be proven by put- inchondensabile ting water in a container of narrow mouth; unless one uses some machine, one cannot put ingiegnjo more (water) than the natural capacity of such container. The contrary is seen in the case of air forced into containers of very narrow mouth. If in them there is some wat­ er, and one turns the container, so that the water be between the orifice of the container and the compressed air, the 'potentia' of such compressed air expels the water from the container with such violence that it penetra- furore tes far into the air, until the air remaining - in the container returns to its natural orig­ inal density. rareta IV-5

But, to return to our _topic, we will say that proposito amonq the movinq bodies of equal gravity, the gravita one with larqer area in its front against the fronte dujditricie ai_r will be the slower. The converse will apply to the body which occupies small area; such body will do the opposite. But one should not do this with such thinness that sottilità the body JLack s_weiqht, because without weight peso li manchi there is no motion through the air.

No local motion through the air is possible lochale unless (the body) has greater or smaller density than the density of such air. And if the adversary would say that the same density auersario of the air compressed in front of the moving chondensata body, creates the same obstacle to the motor, and even more because the motor has contact with more air compressed in front of it be­ cause of impact, like the hand when throws the stone in the air, we can answer here that it is impossible, to the motor to be more or less rapid than the body to which it is join- velocie ed. And never will any part of the acciden­ congiunto tal motion have velocity equal to that of the said motor. And this is proven in the (study of the) accidental motion in which at each degree of motion there are acquired degrees grado of slowness, although the impact of the 'mob- tardità ile' is greater though somewhat remote from remota the driving motor.

And this we see in the arrow pushinq aqainst saetta dellarcho the wood (of the arc), even if the cord push­ es it will all the 'potentia' of the arc it penetrates very little in such wood, contrary to what it does having some velocity. Some moto say that as the arrow moves induces a wave of vnonda IV-6

air before itself; such wave through its own motion does not impede the way to the said arrow.

But this is false because any moving body burdens and makes impedement to its own mot- affaticha impedisce or. Hence the air affected in front of the inonda said arrow is acted upon by the motion of the sinonda arrow and offers little or no help in motion favore to its motor, which must be moved by the same motor; rather, it obstructs and diminishes impedisce the motion of the moving body.

The 'inpeto' generated in the quiescent water produces an effect different from that in the quiescent air. This is proven in that the water is not compressed bv some motion done chondensa under its surface as it does the air which is impacted bv the movinq body. But the parti- cles that fill the water from surface to minutie singhonbra bottom and turn around fillinq the vacuum siraggirano left by the fish which penetrates the water, show us what happens. The movements of such water impact and push such fish, because only that water under water which flows (relative acqua infra laequa to the rest) can exert force. And this is a peso first cause which contributes to increase the chausa primiera motion of its motor (by the water itself?)

CA 301 R (108 v.a) c. 1487-90

Drawings of a siphon and a canal with the letters a - c b - d: IV-7

Any qreat river will be driven up very high si chonducera mountains by means of the siphon. cicognjola

The height 10 'braccia' and the width 8. If the river o d b has a branch up to a and goes ramo down again to the point b, the weight along the line a b will be so much more than that along the line a o, that it will be possible to derive so much water to raise boats up the rubare mountain. IV-7 IV-8

CA 302 R (108 v.b) c. 1490

Marinoni considers this a difficult passage3 full of incongruencies and hesitations. Marinoni's version has been used> hut in a critical analysis of this page it may be very useful to take into account the words and sentences discarded by Leonardo.

Among the damaging actions on the land, T believe that the ruinous floods of the rivers are paramount relative to those of the fire, tengano il principato contrary to the opinion of some, because the destruction of fire ends when there is no voragine more food for its subsistence.

Among the irreparable and danosi furorj surely the floods of (turbulent) destructive turbolentj rivers must be given priority to any other horrible and scaring action, and not the fire. Some want to give to the fire prefer­ ence with respect to the floods. On the contrary, I find that the fire consumes that which feeds it and consumes itself together with its food. (It is true that) the water flow is maintained by the sloping valleys and ends with the last descent of the valley. But the fire is caused by the combustibles notrimento and the water flow by the inclination. The basseza food of fire is discontinuous and discontin- disunito uous and separated is the damage; the fire dies where its food lacks. Instead, the inclination of the valley is continuous and declinatione continuous is the damaqe of the destructive ruinoso water stream, until together with its valley ends at the sea, the universal low point and IV-9

only rest place of the wandering waters of peregrinante the rivers. But what words can I use to vochavoloj describe the abominable and horrifying dama- qes aqainst which there is no human protec- riparo tion. (The river) with its swollen and domi- superbe nant waves ruins the high mountains, tears deripa down the strongest banks, uproots the well disuele established plants, and with its rapacious waves, inturbidated with the cultivated intorbidato lands, takes away all the utmost efforts of poor and overworked farmers. It leaves the valleys denudated and miserable in unleashed vilj poverty.

Amonq the irreparable and damaqinq furies furorj surely the floods of destructive rivers must ruinosi be put first relative to any other horrible and horrifying damage. But with what lang- lingua uage, what words, can I express or tell the nefarious damages, incredible bank destruc- deripametj tion, inexorable rapacity of the floods of rapacious rivers, against which is useless rapacj any human protection. IV -10

CA 303 R (109 r.a. and 109 v.c) c. 1490

Right-hand-side page. Drawing of triangular vessel :

If the water comes out, passes or falls along the vertical through a triangular (cross- section) conduit...of uniform internal area, equale vachuita what cause reduces the shape to the circular retonda proportione one before the end of the flow?

Sketch of flow in a tank with a bottom orif­ ice:

I say that the surface of the wine in the case that the wine goes down through a low orifice is more apt to effect the descent of spirachulo chomada a such wine than any other of its parts. restavrare inchalo The reason is that a body will move more easily if it is included in the less resis­ tant thinqs. That wine at the surface is in ujno superfitiale between the air and the rest of the wine. The wine under the surface is between wine and wine. Hence the surface (layer of) wine which is between a liqht (body) and a heavy legieri evno grieve _one, will more easily move than the second (layer) which is in between heavy and heavy (layers).

Right-hand-side page, first column:

Each motion will follow the direction of its via course along the straight line for as long as the impulse given by its motor will last. violenza fatta IV-10 IV-11

The water (be) in a reservoir or in a canal bottjno of uniform height of its walls. In one of its walls let us have a square cut which is like the empty_spaces one finds in between vachuj one merlon and the other. The water in the merlo canal which comes upon that crenel will move crena towards the crenel. The other (water?) as it leaves the reservoir will be lower than the bottino waters at the right or the left; hence it is necessary that the water at the right go down and passes to the left; and the one at the left passes to the right.

Sketch of a tank tilth the letters r a n - e - s b m:

If vou will make an orifice in the vessel at buso neluasello e the wine which is in a h is put in the middle of the wine n m . ^nd from the bottom of the vessel p s , if you drill the orifice at e, you take away a part of the _sj¿ppp_rt to sostentaculo the line a. b • hence such line bends there, and bv bending it shortens alonq the vertical perpendichulare line; and by shortening the surface of such wine, because it tends to _stay__plane_J, _c_onc_u_rs star piana sochore itself to the emptied place. But because one finds areater velocity in the wine that falls presteza down vertically than in that that concurs to sochore the surface concavity, the concave place chonchauj ta remains concave, because the concurrence is sochorso slower than the fall (of water). disciendere

Sketch of tube tilth 'se po', and a second tube with a e: IV-11 IV-12

If you have a tube full of water, and the canna tube is open below and above; and the lower end is pressed forcefully against the ground so that water does not flow out or leak out. And you will fill the tube with water, and then suddenly you will raise the tube. Then chonsubita presteza you will see the water stay for an instant in the shape it had in the inside of the tube and with almost undetectable rapidity the quasi invisibile presteza water will deform circularly becoming flat and round. And the more perfect the circle will be the more perfect the plane is, more than it was in the tube where it was higher before it became spread around. ispianata

Third column:

Make a plan which is simple and the demon- propositione stration with figures and letters.

If the water that is inside the tube cl c cana stays straight, one should not wonder about diritta that, because it is sustained and held toge­ ther by a body harder than it (the water). But the water, which is set free, is outside the tube, inab, and touches the earth. Why does not bend somewhere since it is not held non si piega

from above? I say not held, because the sostenvta water which is in the tube quickly presses and sustains itself on top of the one that comes out, rather than the one that is out being held by the one inside.

The water that becomes free by falling out the tube and becomes surrounded by air which is less resistant than the tube, does not non si torcie bend because the impact it makes on the earth IV -1 2 IV-13 undoes so quickly the cohesion of the oncom- unj tione ing water that the water as it falls finds itself always without support. Being in this fondamento state, the water does not find where to rest upon, hence it cannot bend, because the mat- torciere erial line which is not comprised between two resistances cannot bend. IV-13 IV-14

CA 303 V (109 v.a) c. 1490

Upper margin, on the central fold:

The element does not weigh in its own element unless it is because of impact. perchusione

Verbiqratia: if you blow air, with a bel- mantacho lows, into the surrounding air the blown (air) will penetrate the air in its way. In this way will water be penetrated by another water falling on it.

To the right, a tube resting on a hand:

When one puts 100 pounds of water in a tube like the one of a blowpipe and one puts un- cierbottana derneath the hand, will the weight of such water (still) act on that hand?

Man lying on the bottom of a tank:

If a man lies on the whole bottom of a water container with ten thousand pounds, will the weight of that water act on that man, or rimarebe not? being the container as shown here a- bove.

To the left, vessel with the letters a - c b:

If the bacj a has on top of it 10,000 ^thou­ baga sand) pounds the bottom b e does not sense them, because the water rests on the bag, and the bag is trying to float up. As this hap- si uole levare in alto pens, the water will overflow (at the top) siuersera and because of this the bottom is not acted non sente peso IV-14 IV-15

by the weight (pressure?) of the water; only by the air below it.

Sketch of vessel with granular (?) material:

If I let the water escape at the point b in fuga the above vessel, I am sure that from the son chiaro buso orifice will not flow other water than that which is above it along the vertical line. The reason is that above b at cl you will see shaping up a small djjnple. (If) one asks if buso the water that flows out is any other than that along the (vertical) line, I say no, because it is easier to flow for the water at the surface which is between the air and the rest of the water than it is for that water which is between two layers of water. Verbi- qratia: if you have a qreat pile of wooden monte di legnje lo_gs, you will see that when you try to bend it all, the upper layer is the first to move.

See (also) the timer in which, when the ver- oriolo tical line is made of red powder and the rest poluere rossa is white, the red pours before the white one.

Two vessels with the letters b - a;

a discharges as much as b :

Two sketches with the letters a - e - b:

Water cannot be compressed, be where it may be, but it can be rarefied when, in this rarefare apparatus, (water) would pour out through e . strumento

And it would not discharge more than given by a b, And if it would yield more it would be because air would enter the vessel. bottino

IV-16

CA 306 V (110 v.b) c. 1495-7

Family of curves¿ jets? paths of projectiles?

CA 310 V (112 v.a) c. 1505-8

*Water covers great part of the earth and veste captures the image of the sun in its surface and thus shines in the universe. It makes itself star in the same way that the other stars become visible to us.

*The image of the sun in the water wave grows inversely to the decrease of such wave's figure as it is farther and farther from the eye.

*If you look at the island surrounded by waves reflecting the image of the sun, it will seem ripiene delli simulacri to you that you see one of the spots of the machule moon surrounded by its (the moon) splendour splendore (or brightness).

*If there are waves in the moon, taking into account that waves do not exist without wind, and that wind does not originate without terrestrial vapor carried by the humid drawn vapori terestij into the air by heat, it follows necessarily that the body of the moon should contain IV-16 IV-17

earth, water, air, and fire having the same movements which have our elements.

And if you say that gravity is nothing else than one element drawn into another it fol- tirato lows that where there are no elements there is no gravity. Hence the moon does not weigh in its own elements and cannot fall from its ono po chadere place.

Sketches with the Sun and the Moon and the letters r a S - p q f

It is impossible that the image of the sun be simulacro seen on the lunar sphere that occupies one half of the body of the moon, unless its surface be full of waves.

The adversary says that there are no waves in aversario the moon but minute, polished protuberances, grobosita mjnute pulite which can capture the image of the sun. This is answered with the fourth of the obser­ vations on the moon, where it is shown that the brightness varies as the waves vary: splendore larger or smaller. IV -1 7 IV-18

CA 311 R (112 r.b) c. 1508

That thing is highest which is most distant from the center of the world, and lowest which is nearest to that center.

The horizontal place is that in which all sito della equalita parts are equally distant from the center of the world.

The adversary says that such place should be spherical, that is curved and not plane. He says also that the plane is that (surface) in which a straight line drawn from any end of stremo it will contain all that line. To this the answer is: that is something to be called rather a concave than a plane surface, be- sito piano cause, if we pour water all around, that water will quickly run toward the middle of the plane, and it will stop there with great depth in the middle and no depth at the sides. Hence the bodies dragged over such stracinati mathematical plane from the middle towards the ends would show an ever increasing fric- confregatione tion as the bodies would approach those boun- daries. And this is against our proposition proposta which begs that the place where the friction vuole occurs be of such a quality that such fric­ tion does not change in 'potentia' in any part of the motion.

- Sketohe8 with Water - Earth - Earth: IV-19

If the Earth would be cut in half, what would the water and the remaining half of the Earth do? IV-19 IV -20

CA 311 V (112 v.b) c. 1508

Drawing illustrating the mathematical and the natural planes:

...Natural plane is that one in which the water poured on its middle does not move unless over a circle with thin and brief motion, effected perpendicularly. Such water per linia perpendiculare will not flow unless it is through an in­ crease of its circle.

A sphere placed on the mathematical plane will only be at rest at the center of such plane, but will be at rest in any point of the natural plane.

An air wave cannot push a ball from the shot- scoppiecto gun into the iron unless the air pushing ferro against the ball is denser than the said pallotta ball. And if that happens the ball is cru- shed between the iron and the air. frange

A drawing illustrating the experiment of the hall going through a number of oheek planes. a 1 2 3 4 5 6 7 10 100 1000 10000 100000 1000000 10000000

8 9 10 11 12 13 100000000 9

Left-hand-side margin: IV-20 IV-21

a is one, which after passing 12 cloths comes tele from a million of millions. And thus in 12 'braccia' distance the bullet pierces 12 cloths, moved by one million of millions degrees of 'potentia* , if set as unity when resta passing through a .

Right-hand-side margin:

Each 'potentia' of impact which is made with­ in water or air, proceeds by propagating in si ua dilatando the form of a sector in the said water or air.

Drawing3 similar to the previous one3 with: 10

In the center:

If the bullet of the shotgun passes through scopietto the wine vessel, the air that pushes it closes up behind. riserrata

In a passes the bullet with 'potentia' one as it moves towards the center of the world. mondo

Two fish and an arc with arrow:

What will do the arrow that is impacted and percossa sosspinta pushed upon.

The string of (really) uniform resistance equal resistentia will never break (at one point only).

Take a hair and have two oxen pulling from it IV-21 IV-22

in opposite directions. You will say that such a hair will break in the middle, but I want it to increase its strength in the mid- piu forte di grosseza die to compensate for weakness there; then such a hair should go into dust all at once. poluere IV -22 IV-23

CA 323 R (117 r.b) c. 1490

It is true that I would not know like them to chome loro arque auotinq authors, but I can aduce with allegare gli altori greater dignity because I can refer to exper­ ience, teacher of teachers. They qo around, maestra ai loro maestri conceited and pompous, dressed up and adorned with the efforts of others, but they do not want to give credit to my work. And they will not recognize in me the inventor; they are not inventors but broadcasters and speak- tronbetti e recitatori ers for the work of others, and they are those who can be discredited. biasimati

CA 327 V (119 v.a) c. 1490

Right-hand-side column:

Many may believe that they can criticize me riprendere with reason, arguing that my proofs are a- qainst the authority of some men which are alturita revered regarding their inexpert judgment. In this they do not consider that my conclu- chose 5j._Q.QS_ are born from simple and plain exper­ ience, which is the true teacher. maestra vera

My reasoning is meant to make you d_i_s_tin^u_ish_ fartj chonossciere between true and false. This makes men ex­ pect with moderation the things that are possible. (I wish) you do not become blinded tu non ti veli by ignorance, because you would despair and become sad when thinqs do not happen. darti malinchonja IV-23 IV-24

Preface

As I see that men before me have worked on preso per loro all the useful and necessary subjects and teme that I will not be able to dwell on t_qjDÌcs_ of materia qreat profit or enjoyment. I will do as the grande vtilita o diletto poor man, who joins at the last moment the fair, who has to take all those things al- fiera ready seen by others, and not wanted but rejected because of their little value. Those goods, disdained and refused by many

J3 uye_rs_, I will put on my weak shoulders and I chompratori will go to the small towns rather than the big cities, and I will give them away, har­ vesting such rewards as are worth of the things I will be distributing.

CA 333 R (121 r.b) c. 1515

By a hand other than Leonardo’s:

The motion of the thunderbolt bends from the saetta dense to the rarified, because it does not find the way of least resistance? because of transito this there are more thunderbolts going up than down. And if the motion goes from up to down, it will often show a bend of the straight path, because the more it goes down the denser will the air be. And frequently, piu grossa the bending is due to the condensation of air flexitudine in front of the thrust of the thunderbolt and resists (its advance). And this is the cause for turning towards more expeditious path. expedito transito IV-25

Where the air is not compressed, it offers condensata less resistance and because of this the air that raises from the bottom of the water to its surface follows a sinuous path because only the water vertically on top of the air exerts pressure, and so much goes the air up peso as the water goes down. But, because the water descends towards the direction in which it acquires weight, per force the air must be bend aside and deviated from its straight i pieghi in disparte motion up until it gets itself in contact with the common air.

By Leonardo’8 hand:

The motion of a thunderbolt, which comes from saetta a cloud, is sinuous because it bends from the flessuoso dense to the rare (air) . Such density is denso al raro caused by the Ijn&etujpsity of the above men­ furore tioned motion. Such motion cannot proceed with the initial straight path, and bends where the path is more expeditious. It fol- spedito lows the new direction until it finds the second obstruction, and following this rule inpedimento continues until its end.

CA 334 R (121 v.a) c. 1515

Above the vertex of the pyramid : IV-25 IV-26

Any pyramid constructed on a square base can be divided into eight pyramids similar to the si risolve given pyramid. al suo tucto

On one side of the pyramid :

If the pyramid constructed on the square base is cut at the middle height by a plane paral- tagliata equidistante lei to the base, it will be divided into two parts which will be in the ratio of seven of one.

Small pyramid with the letters a - d - c b:

(Pyra) mids with _shape_ similar to the entire f igura pyramid. You will say: on the side a b of the pyramid a b et two times two a o makes 4 triangles. And then in e d: two times 4 makes 8 similar pyramids to the orJu^ina_l_ maggiore pyramid o d a .

CA 340 R (c. 123 r.a) c. 1495-7

As the fire is the lightest element it is also of least resistance; thus if it were resistentia possible to take to the maximum height some auantitv of air such air would perforate such preforerebe element (the fire) without ever stopping in the fall until it would reach its sphere (the air would reach the air). Similarly because the air is lighter than water it has less IV-27 resistance. Hence when the water that evapo- lassù vapora rates (and qoes up) reduces to its simple nature (condenses) it descends perforating the air which cannot offer resistance and it returns by the shortest path to its own ele­ ment. Similarly, the earth, heavier than the water, if it is placed at the surface of the water which, being lighter, cannot offer resistance, will go the bottom of the water by the shortest way....

Aristotle says that each thing desires to desidera maintain its own nature.

The gravity, daughter of the motion, like the figliola 'forza' desires to disappear, and because of this each maintains its existence with vio- violentia lence. And if it were possible to make a hole alonq one diameter of the sphere of the pozo Earth from one side to the other and one would let a weight fall in the hole, even if corpo grave such a body would want to stop at the center of the earth, the 'inpeto' would prevent that uieterebe from happening for many years.

To the left of preceding paragraph :

The 'inpeto' which is born from the motion delays in many cases the desire of the thing disiderio that is moving. IV-28

CA 341 R (123 r.b) c. 1510-15

Upper margin:

Why the water and the ball move ìLq w DJLLL* The alia china water moves because of its inequality (in inequaluta location) relative to the center of the world, and the ball moves for the same rea­ son.

CA 342 R (124 r.a) c. 1508

Amona the waters flowing throuqh equal orif- spiraculj ices the one which enters the orifice with greater velocity will have the larger dis- charge. This is what happens in the outlets sara piu ahondante boche close to 'Abbia'.

In waters of equal depth, width and indi- largheza nation that one is faster which is near the surface. This happens because that water, laequa di sopra above, borders with the air which is of small resistance being lighter than water, and the water below is in contact with the earth which offers great resistance because it is at rest and heavier than water. It follows inmobile that the part that is more distant from that bottom is of less resistance than that above, which borders with the air which is light and lieue e mobile mobile. IV-29

Hence the orifices close to the surface of bochelli the water do not lose that quantity of water that. . .

That water is faster that descends with more inclined slope. linea piu obliqua

That orifice pours more water which receives spiraculo the water with greater velocity.

Among the waters which flow through equal orifices in equal times, that one will have spiraculj larger discharge that receives the water with piu ahondante larger velocity.

It is proved in the first of 'De ponderibus' that among the bodies of equal size the ratio grosseza of the velocities and the lengths of their motions is the same as that of their weights. I.e., if there are two bodies of equal size gravi but double weight one than the other and they are allowed to fall in the air at a given instant, I say that because of the double ratio of weights the motion (length) and the moto velocity will also be in double ratio.

Of the weights of equal size the ratios of velocities and weights is the same. simile

CA 347 R (126 r.a) c. 1487-90

Sketch uith the letters A m t n:

How the lines or rays of light do not go through unless the bodies are transparent. diafanj IV-29 IV-30

Sketch with the letters a-tm-brn-do - c :

How the lines of impact propagate through any cholpo passa wall.

Sketch with the letters oc-mba-ntv:

How findinq a hole, there come out many lines foro each weaker than the first a b.

Sketch with the letters ex-ta-c-Sb- o n:

The voice of the echo. boce

Sketches without letters :

How the lines of the magnet and those of the chalamj ta iron pass through the wall, but what is ligh- ferro ter is attracted by what is heavier. e tirato

Being of equal weight, the magnet and the chalamj ta iron attract each other equally. sitjrano

The smell does the same as the impact. odore colpo

CA 348 V (New) c. 1490-2 IV-30 IV-31

The heavier part of any body moving with 'inpeto', will become the guide of its mo- t ion. IV-32

CA 349 R (126 v.a) c. 1487-90

Vessel pouring water with seeds3 with the letters m - f a r:

Here the water flow close to the sida luscita surface and it is asked what part of the surface water will acquire faster or slower motion to provide water to such opening. And to find the rule, put particles that will stay floatinq (in the water), that are uni- a noto form, like some seeds of herbs, and put them erbe in a circle with center at the opening in r m f s and note which one happens to reach first the opening, stop the water and look at bocha the circle.

Cross-section of vessel that pours through two orifices with the letters a - b:

Here it is asked: which of the two tubes canne will give more impact on their target, the obbietto tube a, or the tube b, because in the time that the upper tube pours out a quantity of water, the one below pours 3; hence your ans­ wer.

For equal distances from the opening the impact is the same and this is valid for all flows from an orifice.

Container with perforated bottom:

Here it is asked: if a vessel has the bottom uaso perforated uniformly like a sieve, which of crivello the orifices will pour more water in equal IV-32 IV-33

interval of time. You will proceed this way spatio to experiment and then make a rule: open only one orifice at a time, and weiqh the buso water over a space of fall of one 'braccio', or more or less if you please, and then close the orifice and do the same with the other orifices, closing those already tested. But replenish the vessel with water without im- ristori perchussione pact, so that you do not exert pressure at some place on the bottom of the vessel recei­ ving that water. Be sure that the vessel has always the same weight of water. To achieve this, it is necessary that the vessel that receives the water be separated from that one that pours it out.

Container divided by vertical lines3 with the letters mnfedcba-gh:

If you have a water vessel of uniform width and height, and you divide the width of such water equally as shown in abcdefnm, you will see that each (portion) pushes to become plane. And if you open suddenly the ispianars:L walls of the vessel leaving free all the water, you will see each portion of such water go away from its vertical centerline. perpendiculare The said center line would be the last to come down.

Sketch of container :

Vessel with millet mijglo

If you want to demonstrate what part of the fare pruova water in the vessel is the one that is pour- viene furoj inq out, do the following experiment with the sperientia IV-33 IV-34

vessel with millet. The millet is smooth and lubrico minute. Open and close holes of the vessel and you will see whether or not the plane above in the millet comes down where it is vertically above the opening.

And if you would say that this is not a good experiment because the water is in fact a sperientia material united and continuous while the unjta e continua millet is disunited and discontinuous, I will tell you that I want to use the license which is common to the mathematicians, that as they divide the time in degrees and of a contin­ uous quantity they make a discontinuous one, I will do something similar and make compar­ able millet or fine sand and water. renella

Vessel with the letters deba:

...that the water that pours out from the vessel along a vertical line, is always that one which is vertically above the orifice of the vessel through which such water pours out.

After a very fragmented text.

...And this I say only of the water which do not receive impact from another water which reenters (the container). rientri IV-34 IV-35

CA 350 R (126 v.b) c. 1490-2

Of the sea that surrounds the Earth.

I find the plains of the surface of the Earth to have been in antiquity occupied and cover- abbanticho ed by salt water and the mountains, bones of ossa the Earth, penetrating the earth and going up in the air covered and dressed by much and thick earth. Since then, the many rains and the floods of the rivers, with frequent wash- lauamenti inq downs, have denudated in part the hiqh disspogliati summits of such mountains. By loosing the surrounding earth, the stones became surroun­ ded by air and the earth carried away from those places is...And the earth of the beach- spiaggia es and of the high summits of the mountains has now come down and has elevated the bot­ toms of the seas that circumdated those moun- circhauare tains, and have formed an uncovered plain; and from there in some place they have pushed away the seas.

CA 359 R (130 r.b) c. 1511-13

Left-hand-side margin:

Whether the Ad_icje stays in its place or not. Adice CODEX ATLANTICUS SECTION V V-1

CA 362 R (131 r.b) c. 1508

Right-hand-side column:

Why, when the water falls through air or fire, does it by jerks, flowing with discon- ascosse frusso e tinuity, around its pyramid? The fire does refrusso something similar as it moves upward (in air). Why the motion of fire is slower in cold air than in (warm air)?

Left-hand-side column:

Why the flame does not generate itself if not above a space with smoke, and why does it not impact but its own smoke? This happens be­ cause when flames impact the air they divide themselves into pyramids connected by curved segments; concave, not convex. Something terminj similar happens to air under water.

Gravity and levity are accidental 'potentie', Gravita e lleujta which are generated by one element drawn or tirato pushed into another. sosspinto No (portion of an) element has gravity or levity in its own element.

That thing appears heavier which finds less si dimostra resistance. That thing appears as lighter resistentia which is supported by higher resistance. sosstentata V -1 V-2

CA 363 R (131 v.a) c. 1506-8

Upper part, in red crayon:

If two surfaces are in continuous contact one superfitie with another, without excess or defect, they will be equal.

If a cylinder makes a complete turn over a plane surface it describes an area equal to superfitie its own (lateral) area.

Above the crayon drawing :

The wetted cord becomes shorter, and the chorda wetted cloth becomes lonqer. tela

CA 365 V (132 r.b) c. 1508

Body is what has height, width, length, and _dep_th, and is divisible in all directions. profondità These bodies are of infinite different forms. The visible bodies are of two natures, of visibilij which the first is without shape, or with figura only (some) boundaries which are distinct or terminated; and (if) they exist, they are not termjnati sensible, so that one can hardly see the insensibilj color. V-2 V-3

The second kind of visible bodies is that in which the surface has ends and defines the termjna shape (of the body). The first kind, the one without shape, is that of the subtle or li- sottili quid bodies, which easily diffuse in and mix sinfondano with other bodies, like the mud with water, or like mist or smoke with the element air, nebbia or like the element air with fire or some similar thing. These things have their boun­ dary regions mixed with the neighboring bod­ ies and thus due to the mixture, the boundar­ ies are really confused and indistinguish- insensibili able, thus such bodies find themselves with- out surface because one body penetrates the other. Hence such bodies are said to be bodies without a surface. The second kind of bodies is divided into two sets: transparent and opaque. Transparent bodies are those which on each side show their whole and noth- niente dopo se ochupa inq else is there, like qlass, crystal, water and similar bodies. The second kind of bod­ ies for which the surface shows and deter­ mines their shape, is called opaque. It is agreed that on this kind we will longly dwell, knowing that infinite cases present themselves.

CA 381 V (139 v.b + 139 v.a) c. 1508-10

Two drawings with concentric circles and lines leading to text for them: V-3 V-4

Cut the area of the falcate in half longitu- valuta falcata dinally.

Give different falcates of different curva- curuj ta ture which are equal to (qiven) rectilinear triangles.

CA 384 R (139 v.c-d) c. 1508-10

Right-hand-side 3 from above:

Make the water run into the lock through the concha bottom valve (?). chiusino davello

Make of one piece the gate upstream from the chiusa lock.

Stone slab 3 and 1 - Good

The slab of the reservoir must have a granite labida dellavello box around to prevent the gravel from staying saricco on the lips of the reservoir and obstructing labri dellavello the closure.

Make it lay against the water.

Along the central fold:

Lay the gate against the oncoming water. sostegnj o

Left-hand-side3 near drawing with two concen­ tric circles :

Such ratio exists between current and current as it exists between inclination and inclina- t ion. V-4 V-5

CA 385 b V (140 v.b) c. 1508-10

Drawing with two falcates:

All the triangular falcates of equally uni- piramide falcate formly disform sides, which begin at the disforme circumference and end at the center of their circle are equal, even if they are of differ­ ent length. (This is so) because the space from the center to the circumference is a parallel space and because the straight tran- spatio parallelo sversal segments are equal.

CA 388 R (141 r.c) c. 1508-10

Topographical sketch:

A way of lifting the supply of water. a h is adacquamenti the elevation of the lake of Brivio; h e is piano the course of the Adda river; h d is the course of the canal to be dug. But if I chessi debbe fare extend the level of the lake up to the Tre conducho Corni; i.e., the 4 'miglia', which is the distance h e , i will not do other channel than e f which acquires 'adacquamento' over all the distance f d, i.e. 4 'miglia'. Hence, in this case, one avoids the work of 8 'mig­ lia' of excavation, which is very much diffi- cauamento cult, and it will receive water up to the si adacquerà insino alle feet of the mountains. And if you extend the radici V-5 V-6

lake to h what will result? This will result che farebbe in the lake not beinq wider than the canal. piu largo rozza The canal, flowinq fast, would empty and dry veloce it.

CA 388 V (141 v.b) c. 1508-10

From above:

From the beginning of the canal to the mill. principio navilio

From the beginning of the Brivio canal to the Travaglia mill there are 2794 'trabochi', i.e., 11176 'braccia', which is somewhat more than 3 'miglia' and two thirds. And here I find the canal 57 'braccia' higher than the

skin of the water in the Adda, if one gives pelle two 'oncie' of drop each hundred 'trabochi'. chalo In such a place we plan to place the mouth of sito our canal.

To the rightya drawing of a baseone' : us raise the bottom downstream from this gate sotto by one 'balestrata', to make the water back up and stay without velocity. corso

Thick cloth in between the 'armadura' and the tela grossa water, and the water and the bottom.

Make a concavity in Tre Corni where it ends the wall that contains the water. chiude

Plan view of the works:

Canal to supply water. canal da adacquare V-6 V-7

Travaglia mill - Adda - 'Rochetta a Santa Maria'.

Lake of Lecco backed up at the Tre Corni in the Adda.

Perpetual lock. concha perpetua

Here the gate is of one piece.

Travaglia mill.

When digging the earth for the lock use coun­ ter-weight with water in the vessel. cassa V-7 V-8

CA 398 R (147 r.a) c. 1500 (?)

Upper margin. Flow from a vertical tube;

If water comes out from a full tube, the water will flow down with cross-sectional equale per grosseza uniformity and without twisting, and it will intersegationj be fast in its flow. moto

Water flow of triangular form:

Water of triangular shape. Whoever believes piramidata this makes two errors. First: that the pyramid ends in nothing. Second: that it ne va injente will become so light that does not move any­ more .

Water flow of permanent cross-section spread­ ing on the floor:

Water of uniform cross-section and increase continua of motion (velocity). What happens in this moto case is that at the bottom the velocity would be so high that it would fill up 100 vessels vasi in the time that one pours out above. And this is impossible.

Water of uniform cross-section and velocity. continua grosseza This would give the same impact at the begin­ ning that it gives at the end and would go aaainst the statement that the more the sententia weiaht falls the more heavy and rapid it piv grave becomes. V-8 V-9

Right-hand-side column.

Along margin, a long twisting jet:

Here the weight makes (the water) turn around its vertical. When the tube, or conduit that perpendiculare pours (the water) is of more uniform cross- equale di uachuo section, the pyramid formed by the flowing water will be longer before it reaches the intersection than the water pouring from the tube or conduit which is more pyramidal.

Here arises a doubt concerning the fall of a vn dubio discrete quantity, i.e., of the balls that quantità discreta are let fall at equal time intervals. And this consists in that you want the excesses eccessi to be in a continual arithmetic series. The proporzione continua aritmetrica second ball will exceed the first by a double space of motion; and the hundredth will not exceed a hundredth of the interval. And if you want the excesses to be of continual eccessi geometric proportion and if you would go down a million of miles, you would see the time of the last one not responding to the time of the first with equal intervals.

Central column. Jets of different shape:

The nozzles pouring out water are of three cane kinds; i.e., wide above and not below, wide below and not above, and uniform; and there are two of a mixed type, i.e., one wide in partj cipante the middle and narrow at the ends, and anoth- sottile er wide at the ends and narrow at the middle. V-9 V-10

Experiment sperientia

Put 25 'cerbottana' balls of equal weight in a tube so that they are one above the other cannone vertically in a high place, and release (the perpendiculare balls) with a string while you are down be­ low. However, the motion will prevent you from recognizing the equal spaces. Try. ispati pari

Definition

If a b has accomplished one degree of descent in one degree of time, b c which is faster because of the 'quinta' will have done one degree of motion and somewhat more. And thus grado will do d which is faster than c, and so on piv velce and so forth. But you should know that if you give motion to one of the said balls in each degree of time, still in each degree of time ends the motion of the last one. finj sscie

Left-hand-side column:

If you pour gradually a jug of water, at each sottilmente un bocale point down the fall (of water) w_i_ll_pass_ a sara empiuto jug equal to the one you poured in the same time the first was emptied. Hence everywhere siuoto the water has the same velocity. velocita

Added on the margin:

In equal time anywhere along the fall the pari same quantity of water is passing.

'Settima'. - That water will be of faster piu veloce discienso fall that pours out from more uniform and long tube or channel. V-10 V-11

'Ottava'. - And conversely, (the water) will be slower if it pours out from an uneven and disequale shorter conduit.

You who build mills, before the water goes down, make it flow through a long and uniform channel (or conduit?). chanale

And the water impacting the wheels should rote have square cross-section, i.e., of height figura quadra equal to its width. And although the upper part comes close to the center of the wheel, it would be better than it would spread in asstendessi much width and would impact the wheel far from the center. Because the water would weight little and impact little. And thus it poco poco is true, make it very wide, the water will move little because it will be very thin; and thus (if you?) make it very high and narrow it will be of small effect because it will valitudine impact the wheel near the center.

Container with little balls, and ten little balls in vertical fall, with the letters and numbers a-bl-c2-d4-e8:

If the ten little balls, which are let fall ballotte in 10 deg_rees_ of time, will have double velo­ gradi city at the bottom, it will happen that the little balls, that were released above in the said 10 deqrees of time, would be collected sarebbe racholte in 5 degrees (of time). And this is impos­ sible because one would collect 10 balls before one let go 5. V-11 V-12

CA 398 V (147 v.a) c. 1500 (?)

Along the upper marginfrom right to left:

The waters pushed up by the force of counter- sospinte in alto weights will raise more JiJirou^l^ tubes than in per via di canne the air, because the tubes keep the water together. But the water without the guidance vnj ta of the tube spreads much and becomes divided sisparge into minute particles, almost like mist, when nebbia it is pushed up by a J._a_rger 'potentia'. superchia

The water climbing up, at each step of motion acquires deqrees of broadness and slowness. grossezza tardità

The bellows which throw water up are acted bottinj upon by counterweights of three different kinds, i.e.: heavier or lighter than water, or equal in weight. And, in addition, they are of three shapes; i.e.; wider or narrower than the container, or equal to it.

The different kinds of weights which exert pressure on the containers are 9, i.e.; wider than the container and heavier than the water,

wider and lighter wider and equal narrower and heavier narrower and lighter narrower and equal equal and heavier equal and lighter equal and equal

And in the case that the material of the materia ■ counterweight would be lighter than the wat- V-13

er, one could yet make the counterweight high enough to play the role of a material heavier farebbe ofitio than water. Like saying: a wooden piece is lighter than water, but none the less take a board and put it vertically and it would make in piede ofitio farebbe the effect of a counterweiqht of lead.

*In conclusion, to show you that it is impos- In soma sible to create with some device made by man strumento a movement of water from down up by means of moujmento the descent of another water of weight and height equal to that one that went down....

Up above3 to the right3 second paragraph :

I conclude that, where the water, does not descend much, it takes pyramidal shape, and piramidale while becoming thinner it weighs less since such weight as it impacts it has come down to the place of impact of that thin body. sottile figura

Of motion.

The weight that freely descends in each de- La gravita gree of motion acquires degrees of velocity and weight. a) All the things that push one another will sospingano be of equal motion and continuous. chontigienti over continue

No effect in nature is without reason. Un­ derstand the reason and you do not need the experience. sperienza

Two drawings of containersy along the right- hand-side margin:

I let water fall down from a reservoir from a vaso height of 50 'braccia' in such a way that the V-13 V-14

water filament coming down has a length of 50 filo 'braccia'. I ask: which 'braccio' will con­ tain greater quantity of water, the first or the last.

In addition: I let fall from a height of 50 'braccia' 50 balls at equal intervals of time ballotte i.e., from letting go the first until letting go the second the interval will be one musi- tempo musicale cal time. And do the same from the second to the third ball, continuing in the same way until the last. All this in such a way that before the last hits the ground, the first will be in the air with all the others. I ask: among the intervals of the balls, which is larger, the first or the last.

The air spreads it (the water?) and retards laspagie it; hence the impact is all over the same. I say, anyway, that water which has fallen more gives larger impact only because of being of greater weight all together in the air, and it is sustained only from below and it is not attached above, in fact it is sustained... apichata

I say first, to define the fall, i.e., the quality of the intervals of the balls, that qualità delli intervallj by the ninth of this (section) , in the fall of each ball, when the fall is divided in ballotta equal degrees of equal height, the ball ac­ quires one degree of velocity. Hence this proportion of degrees of velocity will be a continuous arithmetical proportion, because continva arittmetrjcha it develops together with the differences in the velocity.

Hence, I conclude that the said spaces are equal, because they exceed one another with seciedano over superano equal increment, and because of this the acresscimento V-15

water falling from the same height does the same, acquiring, in each degree of motion, one degree of velocity; hence, by arithmetic proportion, the increment occurs degree by degree of water descent, and because of this it is necessary that the water, as it falls more and more, becomes thinner and thinner. sasottigli But immediately another doubt raises itself which relates to the le_s_ser_we_ight of the mancho pesa water as it becomes thin. This is due to the

second of the first which says that the thing seconda del primo that less weighs, less rapid becomes in its descent. Hence that water does not become thin, because if it would, it would become slower; let us say in such part of its fall which is half way, from the beginning. And besides this, if it would not become twice as fast in two units of time, a vessel would be filled in such thin place, which would not be in the beginning; and this would be impossi- nasscimento ble because the water that woiÜd pour above in one hour, would not be transferred to such a place, where it thins out to half, in 2 hours; it would then be necessary that the water would change into smoke (or vapor), or fumo be multiplied in a tortuous manner, a thing varie torture that we do not see. And if you would like to say that the cross-section remains constant, one could respond with the 'seconda del primo' above stated; being, let us say, twice at the end than at the beginning; then two times more of water would appear at the end of the fall than that which pours out above; such thing cannot happen in nature. And if you would like to say that the cross-section grosseza would be uniform and of constant velocity, V-16

then you would negate the second conclusion negerestj of the first, which you have accepted as true.

Now we can conclude something over the fall of the continuous quantity of water from the quantità continva last conclusion derived for the fall of the discrete quantity of the balls which was quantjta discreta according to a continuous arithmetic propor­ tion.

If the water would become thin, after much going its pyramid would end in one point.

Along the left-hand-side margin:

And if we would want to say that the water in its vertical fall in a continuous manner perpendiculare discienso becomes thinner and faster, and that by be­ coming thinner it is lighter and because of the fifth (theorem) (?) cannot be faster, I would say that the water resting on it from di sopra se le apogia above is what pressures it, and you would tell me that being slower cannot push it.

Somewhat below:

It is impossible that the water which moves some device, could ever... strumento

The water that goes down will never raise non leuera maj another water in the same quantity (as its own) from the place it reaches (down below) to the height from which it departed.

I conclude that in a given part of the fall, discienso the water becomes thinner and becomes rapid sasottigli in motion, so that the air splits it and la diujda changes it from a continuous quantity into a V-17

discrete one which the eye cannot distinguish discreta (as such). And thus from there on the fall is discrete. discreto

CA 399 V (147 v.b) c. 1490-1

Upper right-hand-side corner:

* Any body moving with (high) velocity, leaves a trace of its oath. aparisscie della forma

* Any body moving with (high) velocity, dies tignje its path with the similitude of its color.

Below. One of two almost identical notes:

When the thing that is impacted is equal to chosa perchosa the impacting one, it receives from the lat­ ter the impact, the weight and the motion, cholpo and escapes from its place leaving there the impacting thing devoid of any 'potentia'.

Above the preceding text:

(The first ball) does not go further because it finds the ball in first degree of resis- primo grado tance. And thus, the air which is opposite does not bounce back because it does not find comple_te_ resistance, being the impact the intera briefest 'potentia' and the ball is suddenly arrested.... V-18

To the left:

From the diversity of possible (impacts) result several different sounds, which are of sonj highest penetration in all variety of impac­ ting objects.

CA 404 V (150 v.a) c. 1500-3

Texts interspersed between fluvial sketches with: q- c n- b ah-g - x.

It is not the floods but the decrease.... that takes away the floods.

If a. b is more powerful than b c3 then be will fall to the ground and conversely. This andra per terra is impossible because judgment shows that equality of such forces can(not) take place.

Neither does one understand the line nb. be intende is all empty underneath. Hence ensure that sotto the Arno go through n q and come to g.

To the right:

If q jumps in h, h bounces in x and fills up salta risalta g and it will always break through x.

Sketch on the right with the letters p - r:

The 'Mensola' scours at p. guasta

The Arno, because of the many turns it takes, uolte V-18 V-19

becomes of slow flow and raises its bottom and the waters cannot be contained between the banks and overflow them and originate new argnj trabochano rivers, of which the Bisarno will go through p v • hence cut it and it will give you land from there on. V-19 V-20

CA 407 R (151 r.a) c. 1500 (?)

Up above3 from right to left:

The violent natural motion will happen when some device pushes downwards some weight. strumento Such increment stays always the same until acresscjmento the end of the motion, although it is shifted at infinity.

All the natural 'potentie' must be called pyramidal because they have degrees in conti- piramidalj nual proportion inversely to their diminution djmjnvire and also inversely to their augmentation. acresscimento See the weight that in each degree of its free descent acquires degrees in continuous geometric proportion. The force on the lev­ lieve ers behaves similarly.

From left to right. Drawing with balance and lines which diverge and converge:

Accidental motion.

'Sesta' - The weight, as it goes down, be­ comes faster and heavier. grave

Triangle with weight along line from vertex to center of base:

Falling weight. Fallen weight.

Figure with:

Accidental motion. V-20 V-21

Jetj subdivided in volumes with numbers, impacting side of balance :

7-6-5-4-3-2-1

The impact that water in continual fall makes perchussione in the place of percussion, will never be of such 'potentia' as that of a hard body of a material of same density as water. This is so because the weight that impacts first has descended the total height of the fall and traversed ten 'braccia' in falling, while the second falls only 9, the third 8, the fourth, 7, the fifth, 6, the sixth, 5. This happens in such a way that, when the first impacts, the last has not started moving down. And if a solid body falls, the motion of the part duro that impacts is the same as that of the oppo­ site part.

Two little balls falling down:

If you throw, or let fall 2 balls of equal weight and material, the distance between peso e materia them at the beginning of the fall will be observed always along the motion.

Marginal sketch of subdivided parabolic get:

It is necessary that the water that comes out of a tube be of pyramidal shape, even if the canna tube is of uniform cross-section. The reason is that the _aatlULe._of the fall is of unequal qualità velocity; because the water that has fallen more, becomes faster according to the 'sesta' of this (section), and the one which has fallen less does the opposite. Hence if you V-21 V-22

would throw down lead balls at equal time ballotte intervals, they would not be at equal distan­ ces; rather the distances would diminish as one goes up in a decreasing continuous geo- proportione geometra metric proportion.

A similar property would be found in the spaces of a water (jet) although they would seem equal when measured at the beginning, but you would find them increasing in length as one goes down and in thickness (or dia- grossezza meter?) as ones up. This if the air would non li spartissi not split the water. And similarly would behave the water in the rivers of uniform width, depth and straight course.

Sketch of canal feeding and discharging a lake:

If the water a from one side....

But if the water of uniform canals enters and comes out of lakes, the motion will be such that as much flows out at the end of the feedinq channel as it comes in at the beqin- canale che mette ninq of the discharqinq channel. Otherwise canale (che) versa the water in the lake would stronqly increase forte or decrease.

The water that falls will raise so much more weight than its own the more is the weight or force of the 'potentia' of its impact; and the thinner the higher it is ( its origin ?). But one must subtract (from that auan- issbattere tity ?) the amount which is lost of the 'po­ tential' of the device, due to the friction confregatione of its shafts as it is shown in the ninth. nona V-22 V-23

CA 407 V (c. 151 v.a) c. 1500 (?)

Right-hand-side margin3 with the letters q - o m - o p n:

Experiment to determine the proportions of the intervals in the fall of uniform and eaual weights. Set vertically the board rn n, perpendiculare which is covered with wo^l-cJ_i£pi_nq_s. Con­ terra di cimatura nect to this board another one o p by means of hinges forming like a book which can be closed suddenly by means of two strings, as you can see. At the top of the covered board put the bottom of a 'cerbottana' which should pie be closed below and full of balls of equal pie weight and shape. Then fasten the 'cerbot- ferma tana' and the covered board and open the 'cerbottana' and when you see the first ball about half way along the board, let suddenly go the counterweight and the two boards will close and the balls, which were falling along the covered board, will be fixed on it. Then si ficheranno inessa you will be able to measure the distribution proportione of the spaces separating the balls.

And if vou want to visualize the fall of vedere water make a similitude with millet cominq mjglio out from a hopper, and weigh it 'braccio per tramoggia braccio' , and you will see which 'braccio' does not catch anything.

And if you would like to experiment with a continuum quantity, no substance made liquid quantità continva with fire would be good, because the first part will cool down and freeze when the last is still liquid. V-23 V-24

If you want to make this JL^SJLr the 'cerbot­ prova tana' is not good, because that baJLl that ballotta would be the last would come from a higher point than the first, and would have a higher velocity. This same method would be qood for modo an experiment with two weights, one of double weight than the other, and you will see if the lighter stays at half way when the heav­ ier hits the ground. tocha terra

Two objects, one rectangular and one round:

Equal in weiqht. I want to see if the lenqth lungeza of one (of the bodies) obstructs the motion because it has much contact (with the air?).

Two long objects, one longer than the other:

When you let fall the two weights, one twice the weight of the other, put them with the lower end at the same height, and put at the middle of the fall of the lighter one board and another at the end of the fall of the heavier. Then put your ear close to the sta collo orechio middle of the two boards and with the string in hand release the two weights and listen to the sound of the two falling bodies when they impact the boards to check whether there _i_s_a vnjto o no unison or not; and if not, see which is the one to impact first, and derive your rules. fa le tue regole

Two objects suspended from a pulley:

One weight is double of the other, and of the same material and cross-section, but only (?) of double length and gravity. gravita V-24 V-25

Two objects: one wide 3 the other narrow:

One weight is twice the other, and I can make the heavier J_ess rajpid than the lighter, but men veloce - another person could say - a pound of lead will fall much faster than one of cotton banbaga wool.

Along the left-hand-side margin:

You who say that when one lets fall an oxen bo vivo alive of weight and size equal to a given magnj tudine similar body, the oxen will fall slower than the body, because the body is cold while the oxen is warm: I ask you why when you weighed the two bodies the oxen did not appear ligh­ ter, being warm as it was, than the counter- essendo chaldo weight you put in the balance? If you find the warm and the cold bodies in equilibrium in the balance even if they are of the same bi lande size they will be equal in their fall. V-25 V-26

CA 409 R (152 r.a) c. 1508-10

Upper margin> at the center. Below> figures with spirals:

The circle is a parallel space included bet­ spatio paralello ween the circumference and the center.

And all the triangles uniformly disform, uniformemente disforme formed on the same base and ending in the center of such circle, even if they are of infinite variety in length, will always be of the same area. capacita

CA 411 R (152 ?.?) c. 1508-10

Control-volume type of sketch:

The thing that moves acguires as much space acquissta as it abandons. ne perde

The surface a b moves into a b e c and leaves d equal to d e c t and the (area) a remains common and double. See the proof that says: proua if from two equal things you take equal parts, the remainder is equal, etc. el rimanente

Small drawing with concentric circles and a falcate :

I want to divide this circle in many equal V-26 V-27

parts without using other things that circum- operare ferences. And I will use roots of 1, of 2, of 3 and of (4). Then the falcates will be falcate similarly divided in egual parts along their length.

CA 412 R (153 r.a. + 153 v.c) c. 1495-6

First column3 left half of folio:

If all the bottom of the sea were covered by men laying on it, those men would sustain (or a diaciere sostenerebono support) all the water element, (even) if those men would be with a mile of water on adosso top of them; because if the whole sea is all sustained by the bottom, each part (fraction) of the bottom sustains the part of the water. V-27 V-28

CA 412 V (153 v.b + 153 v.a) c. 1495-6

Right-hand-side page, upper margin:

(Show by) examples that the said stone could not stop at the center of the world due to its 'inpeto'.

Upper right-hand-side corner:

(If the) bodies were perfectly spherical and of uniform density they would have a sinqle materia center; but this seems impossible because matter is nonuniform. disequale

Decide whether the place of the common center difinjsci pulls toward it the heavy bodies while the corpi gravj liqht parts drive them away or not. li scacciano onno

First column of left-hand-side page:

The common center and the center of the Earth dentro comune are not the same. On the contrary, they are quite dissimilar and of different nature, disimjli e varj di natura because the common center does not change unless the place (occupied) by the air and the fire changes, and the center of the Earth is unstable and in continual change. (The instabile latter center) changes as many times as the winds displace the water of the seas by means per londatione of waves which cover or uncover the sea bed. Hence, by taking part of the weight of the Earth and displacing it to another place, the center moves away from the place where the weight is missing, and closer to the place sapropinqua where weight is increased. And thus, the V-29

center of the Earth changes around the common center as many times as the weight of the waters changes upon their bed. And if it were possible to remove the Earth from its place and put it away in such a way as to e tirarla in disparte have the common center well outside the Earth, and then one would let free a stone in that air which would be in between the common center and the center of the Earth, such stone, without any doubt, would fall toward the common center following the most direct uja piv brieue path.

Sketch with the letters c b a:

The weight that reaches the center loses its gravity. Let a be the center of the ele- graveza ments, i.e., the common center, and c the cientro cumune center of the world. Let b be the place where the stone is released in the air at equal distance from the two centers. I say that a will descend to the common center, and it will not go up to the center of the Earth. non si leuera in alto

*The center of the universe is not the center cientro dellunjuerso of any element, because the elements exper­ ience continual changes of place due to the continual celestial influences which produce continual revolutions and various accidents revolutione (on the elements). The earth, in great part covered by the water sphere, forms a unique body with such water and floats like a (can- sta sosspesa on) ball amidst the air. The center of the water sphere is not the center of its grav­ ity, and its gravity center is not the center V-29 V-30

of gravity of the earth, but in fact is quite far away; hence the water surface is not equidistant from its center....

• • • • • •

Comments :

There is a long discussion of the centers and of the different spheres, and about displace­ ment of elements to different spheres and their motion thereof. However, it does not seem justified to reproduce all these texts which are repetitive and do not say anything new relative to other texts. V-31

CA 413 R-V (153 r.b-c +v.e) c. 1513

According to Marinoni3 the incomplete Latin text in this folio comes from "De insidenti- bus in húmido" of Archimedes. Marinoni doubts that Leonardo could ever have been able to read this work because of its special difficulties (e»g»3 abridgements) and the general lack, of experience of Leonardo in reading Latin. All this is important because it comes in support of the notion that Leo­ nardo approached hydrostatics in his own way. Such way was entirely different from that of Archimedes. V-32

CA 415 R (153 v.d) c. 1493-5

• • • • • •

Red crayon. Water jet impact on balance 3 with om-c-n-ba:

Experiment and formulate rule for the differ- Pruova enee between the impact of water on water, cholpo and that of water on something hard. And chosa dura take well into account this: that even if water is falling in water and the latter is making room to the impact, the percussion that throws open the water receiving the che fa aprire impact, could exert on the container the same action that is received by the impacted water. And this would be the same as if the falling water would impact on _some_thing_hard_ chosa dura and capable of resisting the impact.

If you want to make a rule on how much is the 'ootentia' of the impact which is simultan- cholpo achompagnjato eous with the weight of the falling water, weigh first in the balance a the weight and peso e cholpo impact of the water c b on the balance b ; then weigh the water without the impact, i.e., the water which is between n c. Know­ ing the weight of the former, you can know how much was the impact. Now it remains to weigh the thrust that was applied by the spingere water above m o, which will be weighed if you touch with the balance the orifice n, and buso then weigh the water which is in the tube c n without the water which is below m o. V-32 V-33

CA 418a R (155 r.b) c. 1515

Doubt Dubitatione

A doubt comes up here : was the deluqe, at dubbio diluvio the times of Noah, universal or not. And it Noe seems not due to the following reasons. In the Bible we find that the said deluge con­ sisted of 40 days and 40 nights of continual and universal rain, and the rain raised the water ten cubits above the hiqhest mountain ghomjtj of the world. And if really the rain was universal, it should have covered completely vesti our Earth in the shape of a sphere. Now the spherical surface is equally distant from its center; then the sphere of the water being in such situation cannot move, because water does not move it it does not qo down. Hence non disciende how could that water go away if it could not move. And how could the water d_i_sapp_ea_r si parti without going up? And here the natural explanations are lacking and to solve the doubt a miracle is needed unless we say that dobitatione that water was evaporated by the heat of the sun.

CA 418b R (155 r.c) c. 1515

One could see people who with great care were stocking victuals in different kinds of boats aparechiavan uettovaglia made very simply because of the u_rgenc_y_. brevissimj neciessita V-34

The brightness of the waves did not appear in lustri those places where the dark rains and clouds tenebrose were reflected.

But where there were reflections of the flashes of the thunderbolts, one could see as vampi cieleste saette many bright spots, due to images of the lustri flashes, as there were waves which could give reflections for the eyes of the people cichusstanti around.

The number of images of the flashes of the s imu1 a c r i vampi thunderbolts was larqer the larqer the dis- saette tance from the waves to the onlookers' eyes.

And thus the number of images diminished as the eyes of the onlookers were closer (to the waves); as it is proven where the brju^htn_e_ss splendore of the moon has been explained, or where the (questions) of the reflections of the rays of the sun at the horizon in the sea, and of the orizzonte marictimo eye receiving such reflections while is far away from the said sea, (are discussed).

CA 422 R (156 r.a) c. 1515-16

Upper left-hand-side corner:

Of the rivers and their courses

Among the straight rivers generated in the same quality of land with the same quantity terreno of water same width, length, depth and inclination of the course (slope) the oldest obbliquita will be the slowest. V-34 V-35

Proof. Of the straight rivers, that one will become more sinuous which is the oldest, and tortuoso of those sinuous rivers the slowest will be the one which becomes longest.

12. - Of the waters which descend from a given altitude to another given altitude, those will be slowest which move along the longest course. chamjno

Of the rivers which begin (at different times?) that will be the slowest which is the oldest. And this originates in that the course (of the river) acquires continually more length due to its meandering. And the torture cause is explained above in 12.

CA 429 R (159 r.c) c. 1515

Up, center of page:

*The air that runs to generate the flame, impacting on it and bouncing on itself, be- refrette comes compressed and becomes visible in a blue shade. In between such visible air and azuregiando aria the compressed flame, on the sides of the visibile flame, there is a very thin air because it is sottilissima the one that comes in contact with the lij3_ht lume and bounces back. In its reflection, meets the other oncoming air and impacts it, and it re-bounces back with continuous rotation and risosspinta indirieto reuolutione riuoltamento turning over. V-36

CA 432 R R (160 r.b) c. 1515-16

Of transformation trasmutatione

Of the transformation of rectilinear surfaces into curvilinear surfaces.

Of the transformation of curvilinear surfaces into rectilinear surfaces.

Of the transformation of curvilinear surfaces into other curvilinear surfaces.

Drawing of a tree:

When a wind hits a tree the backside of the pianta branches is the one facinq the wind, and one si mostra branch offers support to the other. sappoggia V-36 V-37

CA 433 R (160 v.a) c. 1515

Right-hand-side column:

Very great rivers flow under the ground. terra

The rains erode more the feet of the moun- radici tains than their summits. And this, due to cime two causes. The first is that impact of the rain falling from the same elevation is more powerful at the base than at the summit of the mountains, because of the '7a' which says: that weight becomes faster that des- grave cends more through the air, and as it becomes faster it becomes also heavier. Hence, since there is more distance from the base of the mountain to the cloud than from the summit to the cloud, the rain is more heavy and power- piu grave ful when hitting the base than the summit of the same mountain. Therefore it erodes less chonsuma when the fall is less.

The second cause is that there is more quan- maggior somma tity; of water falling on the lower half of the mountain than on the upper half. And thus we have achieved our goal. nosstro intento

The valleys are widened as the time passes. The valleys do not grow much in depth because the earth that the rain gives to the valley terren is nearly what the river takes away, in some mena via places more an in some places less.

On the margin:

More earth leaves the rivers close to popu- terreno lated areas than where there are no humans, spetie vmana because in such places mountains and hills V-38

are subject to works and the rains carry away the land that is worked upon more easily than the hard soil covered by wild grass. teren duri gremignja

The water of the rivers comes from the clouds and not from the sea.

The rubbing of one stone with the other, as confreghatione the rivers flow, rounds the angles of the stones.

Back to the column:

The stones are arranged by the flow of the conpossti rivers.

The stones are arranged in layers, or in falde steps depending on the load carried by the gradi turbolenze flow of the river.

There are no stones where there was no sea or lake.

The pebbles are created by the rivers flow ghiare and at the end are eroded. consumate

The pebbles are smaller the closer the river that generates them is to the sea.

The stone is harder the farther it is (able to go) from its place of origin. basa

The streaks in the stones are breaks made at vene roture dalle the layers when they lost their humidity and dried up; then the cracks were filled with crepature finer material.

The waters laden with fine and unnoticeable turbidity generate (sedimentary) stones at turbolentia the places where they become slow. V-39

Left-hand-side column:

All the mediterranean seas and the golfs of such seas are made by the rivers which pour into the sea.

The mountains are made by the flow of the rivers. chorsi de fiumj

The mountains are undone by the rains and by the rivers.

In their upper parts the mountains are more durable and more permanent (since there) they etternj premanenti are covered with snow all winter. uernata

The base of the mountain is continually dim- resstringano inished.

The mountains become continually more pointed piv achutj

The rivers constantly lower their beds exclu­ ding those places where they are choked up ringhorghati (throttled) because there they do the oppo­ site.

The lakes up in the mountains are originated by the debris of those mountains which obs- ruine truct the valleys.

The destruction of lakes originates from their rivers which erode the banks or the argine mountains which generate the lake. gieneratore

The mediterranean seas decrease continually. mari mediterranj

The origin or the mediterranean seas is the debris of mountains taken away by the rivers. ruine

The valleys continually increase both in width and in length. V-40

CA 434 R (161 r.a) c. 1505

If the motion of the winq that pushes the air prieme will not be faster than the _escap_e_ of the air f ugha under pressure, the air will not be compres- non si condenserà sed_ under the wing and consequently, the bird will not sustain itself on the air.

That part of the air has motion more similar to the motion of the winq that pushes it prienme which is closer to the said wing. And that portion of the air will be more quiescent ferma that is farther away from the wing.

That portion of the air is more compressed condensa which is closer to the wing that pushes it. prieme

The air is denser near the water and near magore grossezza cold regions. In the middle is purer. piu pura

The air of a cold region does not offer re­ sistance to the motion of the birds unless they accumulate much air under them. abraccassi

Drawing of a flying bird:

The air is capable of compression and rare- codensarsi e rarefarsi faction.

The bird is a device which operates according esstrimento to mathematical laws. Man is in power of making such a device with all its motions, but not with enough 'potentia'... . V-40 V-41

CA 434 V (161 v.a) c. 1505

'Inpeto' is a 'potentia' applied by the motor motore to the movinq body. And this 'potentia' is mobile the cause of further motion of the body when it separates from the motor.

That can be seen when the boat towed through the water becomes separated from the 'poten- tia' of its motor: the boat continues to move over some space. And the bird, after beating the wings upon the jci^i^re_ssed_ air, a battute chondensata can move a long way without further beating of the wings, carried by the 'inpeto'. V-42

CA 436 R (162 r.a) c. 1515

Right-hand-side column:

Of three accidents (or phenomena?) Wind, rain accidenti and hail.

That oart of the body makes more way which piu strada draws closer to the cold.

* The cloud freezes above and below and on the sides but not in the middle because the heat acts more upward than adopera laterally. Hence....

* The upper part of the cloud freezes but not the lower part, the sides and the middle, because the heat acts more upward than laterally and more on the sides than downward.

That thinq becomes liqhter than is more heat- piu sirisschalda ed_._ And the one that is cooled becomes heav­ ier.

That thing which is compre_s_sed becomes hot. chondensa

That thinq qets hotter that is closer to heat che piu sappressa (sources?) The converse follows: that thing becomes less hot that it is farther from the heat.

The cloud driven by the heat contained in it, portato toward the fire sphere, finds the cold region of the air which freezes the cloud periphery di fori but not its interior. Freezing of the inter­ ior is prevented by the heat which raises the cloud. Hence three accidents (phenomena) are accidenti qenerated, of which the first is the exhala- esalatione del umido tion (flowing out) of the humidity which, V-43

because of the heat, disunites (breaks up) si dissgregha and evaporates and generates a furious wind. The second is the rain, which originates because of the coalescence of the particles congreghatione of the humidity which has evaporated. The particles with high velocity due to the heat, impact the slow particles of the region of the cloud which become cold. Particles stick together and acquire weight and fall down to earth in the form of big drops. And toward the periphery of the cloud, the particles of the humidity freeze continually forming glo­ globbulentie bules which, because of the extreme cold, cannot disgregate and rush to the place which conchorra generates the sphericity of the drop. The hail stones are formed of many of those glob- glandine ules which come together, etc.

CA 437 R (162 r.b) c. 1507-8

•

Axioms Conceptione

The circumscribed thinq is always less than circhunscripta the one that circumscribes it.

If there are two surfaces which are super­ posed and one neither exceeds nor is exceeded by other, they are without doubt, equal.

Due to the first (axiom) , the second is not general, because, when the surface of the cylinder is surrounded by water or air, the two surfaces are not equal. The adversary laversario says that the surface of the cylinder and the V-44

surface of the water around are the same thing because the cylinder ends where the water ends (on the other side) and the water ends where the cylinder ends, and between one end and the other nothing exists. The answer to this is that the end of the cylinder is of the substance of such cylinder, and the end of the water is of the substance of such water. Hence one concludes that there are two surfaces, and one surrounds the other. And because of the first axiom one is larger conceptione than the other.

CA 441 R (163 r.b) c. 1515

Upper part:

It is the same to iump on a qreat sheaf of gransscio straw or wool, or to descend together with the sheaf of straw or the bale of wool. But fasscio take with you a branch or something else that ramo is light, in your hands, so that your fall be with the head up and not down, because for any body which falls the heavier part becomes the guide of the motion. guida V-44 V-45

CA 449 R (165 v.a) c. 1500-3

Lower half, to the right:

* You, who said that when dropping two bodies in 'sexquialtera' ratio, the time pesi of fall would be in 'dupla' ratio, you were wrong when you wanted to subtract the isbattere air resistance, and you really showed that you did not remember that the weights in the 'sexquialtera' ratio were so because they were found to be in such ratio when they were weighed.

To the left:

* Straiqht the motion of the motor and on a motore circle the moving body. mobile

* Circular (motion) for the motor and straight for the moving body.

* Straight the motor and the moved body: when the horse pulls the boat. V-46

CA 450a R (165 v.b) c. 1515

After turning folio up-side-down. Below, to the right: Sketches of the Earth with a - b and a - d c - b: a moves through b d, and the motion of d is always upward, and lasts 12 hours.

To the right, drawing of the Earth with a tide, and maximum - minimum.

Where there is greater quantity of water, there is greater flux and reflux; the oppo­ site occurs in narrow waters. strette

The cold aggregates; the heat disgregates. congregha dissgregha

See if the sea is in the increasing peak when somma the Moon is in the middle of your hemisphere.

.'Ò. CA 450c R (165 v.c) c. 1505-8

Anv material medium obstructs the motion. impedissce The medium through which a thing moves, of­ fers more resistance the heavier it is. piu grave

Greater support is provided by the heavier thing. In what has (more) d_epth_, the descent profondo is retarded. piu tardo V-46 V-47

If an object of non-uniform distribution of weight is pushed (by a fluid?), the heavier part goes ahead. This means that the heavier part becomes guide of the motion. si fa guida